Nonlinear And Parametric Phenomena: Theory And Applications In Radiophysical And Mechanical Systems (World Scientific Series on Nonlinear Science, Series a)

)j

u>m 2

r

•

vfjqoLo

3.2.S. Forced oscillations The solution of Eqs.(3.54) can be represented as yx = Rcosf3r = Rcos<£>, — = - / ? i ? s i n * y2 = Rsin/3r = i?sin*,

'

dyl

-p-=/?iJcos* or

( 3 - 57 )

where the differential equations for the establishment of the amplitude R and the phase ip of the oscillations are: 2/3-7- = - - / dr n Jo = ( - ^ i

-Pi [yi,y2,2/1,2/2, r] sin*c?* + 2fc2/o/i + 3fc3i?/i - Um amp) B'o - ^ - , (3.58)

d* a [2* 2/3-— = - - / Fi[yi,i/2,2/i,2/2,r]cos*d*

= -uR - ^-IoBi + k2[llB[ + hI2Bl\ + h[I30B{ + ZhhBl] - Um c o s ^ + SoBi1, J_ _ 1

1

(3.58) takes into account the following equalities, obtained under condition (3.57): H(yi) = So + Si cos * + S2 cos 2 * + . . . , /2G/2) =Io~Ii

s i n * + h sin2* + . . . ,

Gi(i/i) = Bi + B[ cos * + B{ cos 2 * + . . . , G 2 (y 2 ) = Bl1 - S [ 7 sin * + 5 7 / sin 2 * + . . . An analogous procedure is applied to obtain — for the circuit of type I

dR R „ 2 — = - — - bi - Umsin(p dr Q

;

(3.59)

Nonlinear resonance in radiophysical systems

171

— for the circuit of type II 2 — = ~^- + 2k2loh + 3fc3/02/i + Um cos^> dT

(3.60)

9 E

The amplitude-frequency and phase-frequency characteristics of the complex oscillating circuit are derived from (3.58)

v=v« ± U(UmBl)*- ^~-^I1+2k2I0I1+U3P0I?j 1

/ x ( - Q - J l + 2kM

^=arcts^I w h e r e VQ = 2 ( " - " » • » ) , ^

\ + 3fc3/ 0 J l ) B0 -

I^BT^R =

1

v.

5o-^|^]

(3-61) S*B"

'

= - ^ . ( ^ 0 " - 1) + ~

\-~IoB[

woo y/LCo ^oo -K (_ y s +Jc 2 [/^B( + hI2B[] + h[llB{ + ZIohB^] + SgBl1} is the equation of the so called ,,skeleton curve" [51]. The condition for the ,,linearization" of the amplitude-frequency characteristics is (o2 + &2) + 6 i 7F"o" + T + T ~ J -R = o-7^r = 2 c(/f [ woo VVE ^ ^ ^ /J In this case the condition for the absolute stability of the complex resonance system is

(^*)(I+4?)+(^+-'+"¥)('+^)>» The analysis of these inequalities shows that the lack of a slope on the skeleton curve I —— = 0 I is a sufficient condition for the absolute stability of the stationary \dR / oscillations. An analogous procedure is applied to obtain

172

Nonlinear and parametric phenomena: theory and applications — for the circuit of type I _1_

S^

.. = ,., ± / ^ ) ' - ( i + §)2, ,= — tg |lf, <«2) where i/Os =

(5^ - 1) +

—

;

— for the circuit of type II

•»'" "£<*'" :) * i f e ) ' - 5 (&-*'•'•-*""•)' ^f^V^-D V = arctg

x

^

(3-63) ;-.

To determine the area of bistable work - for example of a circuit of type I, equation (3.62) is written in an implicit form

F(R, uo) = (~ + S1)2 [ML+^lM _ VoR _ ^LR{Bi W

/

L

W

^oo

_J

J

2

_ Vl

= 0.

The boundaries of the area with bifurcational instability is determined by the condition dv0 _ dF(R,vo)ldR _ dF(R,v0) ~dR ~ dF{R,vo)/dv dR ~ ' After certain transformations the following parabola, delineating the area of bifurcational instability, is obtained

U { Q - l/0){2blb3

+b

*)Q+3{QJ

+

^--

h2V°

(3.64)

where the coefficients bn (n = 1,2,3,...) are obtained from the characteristic of the nonlinear capacitance Cd(q) by using the method presented in [185].

Nonlinear resonance in radiophysical systems

173

Fig. 3.21. Theoretical amplitude-frequency characteristic of a complex oscillating circuit (continuous curves 1, 2, 3) and types I and II oscillating circuits (dotted curves I1, 21, I11, 2n). Curves 41 and 411 delineate the bifurcational area of instability

Fig. 3.22. Experimental volt-ampere characteristic o f a p - n - p - n structure and dependence of its natural differential inductance on the regime current Fig.3.21 shows the theoretical amplitude-frequency characteristics of the complex oscillating circuit (curves 1, 2, 3) and of the circuits of types I and II (respectively curves I1,2T,& and l / 7 , 2 7 / , 3 7 / ), calculated according to formulae (3.61), (3.62) and (3.63) for three values of the exciting voltage. The following elements have been taken into account during the computation: the real characteristics of the nonlinear elements used in the experiment, a quadratic law governing the change in the differential capacitance of the diode with Co =430 pF and the

174

Nonlinear and parametric phenomena: theory and applications

characteristics of the p—n—p—n structure R(i) and L(ty) for iZ/j=l,5 kfi and 2,7 kfi (See Fig.3.22); — = — = 0,02, R = 3,0 Mfi and 0,33 Mft. It is clear that it is possible ,,to straighten" the resonance curves in such a system, i.e. to eliminate the area of bifurcational instability. The parabola (3.64) describing the area of bifurcational instability of the circuit of type I is given by the dash-line curve 4 7 in Fig.3.21.

Fig. 3.23. Experimental amplitude-frequency characteristic of a complex oscillating circuit (continuous curves 1 and 2) and of types I and II oscillating circuits (dotted curves I7 and I11) Fig.3.23 presents the respective experimental amplitude-frequency characteristics for the parameters given above. There is no area of bifurcational instability on the resonance curve of the complicated oscillating circuit, whose type is close to that of a linear oscillating circuit. 3.2.4- Parametric resonance The differential equations for parametric resonance in a complex oscillating circuit (to ~ 2o>o) are:

2/3^

= (--^-h + 2k2hh + 3*3^-0 Bo + VmB[ [1

+ 2/3^

(|g)] siw -^[ 1+ (|)]f» s ,-^, (3.65)

= -vR - -^-IoBl + k2[llB[ + hI2Bl\

dr

QY,

\

/Bn\2

+ kz [llB[ + SIohhBi] + UmB[ 1 + \-±jj \

/-B J \ 2 1 Bu

cos v

Nonlinear resonance in radiophysical systems

175

The amplitude-frequency characteristics for this mode are presented as

—^44 + (f)T + (WHf)T)"~ +^ ^ \ ,

-\(-^I1+2k2IoIl+3k3III1^BIo

(3.66)

where Voh

= -—{BlB» - 1) - -i-Joflf + | [ / X + hI2Bl\ + ^{llB[ + ZhhhBl}. K

For the resonance of the third subharmonic (w ~ 3to0) one obtains CD/J

I

f / 2 7 \ 2 F'i R7 / R 7 \ 2

^+[if(-l|)]>[(-^ +^

7? 7 / R J / l 2

+

^)B«-^]2

(3.67) An analogous procedure can be applied to examine the parametric phenomena in oscillating circuits of types I and II. For example, the amplitude-frequency characteristic of the parametric resonance and the third subharmonic resonance for the circuit of type I is obtained in the following form: — for w ~ 2wo

l(ZUmBl\2

(ISA2

„ _

— for u ~ 3a>o

i * =i "« ± >|[4xU>)J where i/oc =

(B'o - 1) + Woo

ipr^ V-K

"U+^J'

(3-69)

.

Figs.3.24 and 3.25 show the theoretical (Eqs.(3.66) and (3.68)) and the experimental amplitude-frequency characteristics of the parametric resonance in

176

Nonlinear and parametric phenomena: theory and applications

Fig. 3.24. Theoretical amplitude-frequency characteristics of parametric resonance in a complex oscillating circuit (continuous curves 1, 2) and in types I and II oscillating circuits (dotted curves I 7 , 27 and I 7 ', 277)

Fig. 3.25. Experimental amplitude-frequency characteristics of parametric resonance in a complex oscillating circuit (continuous curve 1) and in types I and II oscillating circuits (dotted curves I7 and I77) a complicated oscillating circuit (curves 1 and 2) and in circuits of types I and II (respectively curves I 7 , 21 and I11, 2n). Equations (3.66) and (3.68) yield formulas for the critical amplitudes necessary for the excitement of parametric resonance: — for the complicated oscillating circuit

PL =

=

^

Si [1 + U v J +V

^~2

—^,

(3-70)

B» ) 1 + U v J

— for the circuit of type I (3.71)

Nonlinear resonance in radiophysical systems

177

The dependencies (3.70) and (3.71) on the detuning v$ are shown in Fig.3.26 (for the complicated oscillating circuit - solid-line curve, for the circuit of type I - dashed line). The conclusion is that the excitation threshold of parametric oscillations in a complicated oscillating circuit can be considerably lower than in a circuit possessing one nonlinear reactive element. This fact can be explained by the improved conditions for parametric energy accumulation in the circuit of two parametric elements. The best conditions for parametric energy input are realized for equal depth of the capacitive and inductive parametric element modulation. If, \

nl

nil

/

•fc>0

-°0

\UmcrJ

TTI

for example, B& = B» and - } = - ^ = 0,8, then ( ^

2

) ~ 5Bf.

Fig. 3.26. Theoretical dependencies of the critical amplitudes of parametric excitation on the detuning for a complex oscillating circuit (continuous line) and type I circuit (dotted line) The analytical expressions obtained provide opportunities to examine the influence of each element on the resonance and parametric phenomena in a complicated oscillating circuit. The complicated oscillating circuit considered can be applied in various amplifiers, generators and convertors, active quality filters. Their construction using microelectronic technology is of great importance. The analytical approach outlined above can be used to examine the characteristics of any device (particularly those possessing the resonance systems listed in the beginning of the section), which can be presented in the form of a complicated oscillating circuit with two nonlinear reactive elements of antipodal nature.

3.3. Generalized modified method of complex amplitudes for analyzing processes in nonlinear oscillating systems Consider a nonlinear oscillating system of second order described by two nonlinear differential equations, which can be presented in the following vector form: —x = A(x)x + f(r),

(3.72)

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Nonlinear and parametric phenomena: theory and applications

A(x) = {a o -(x)}, •,; = 1,2,

x

=[^]'

(3.73)

f(r)= H f ^ c o s ^ r + ^t)L -I Jb=o

System (3.72) can be solved by using the method of successive approximations presented below, where, for the sake of expediency, it will be rewritten as

jU(" = A^'- 1 V > + f(r),

(3.74)

x"" 1 ), x ^ are the solutions of system (3.72), respectively in the (/— l)-th and l-th. approximation. Initially we shall determine x^'' assuming that x''" 1 ) is given. During the analysis we shall use a special version of the method of complex amplitudes. Some major transformations related to this method were explained in section 1.2, while its overall presentation is given in section 5.1 of the book. It follows from some general considerations that the solutions concerning approximations of arbitrary order can be presented in the form x('-1)=

oo

oo

£ x('-1)COs(^r + ^'- 1) )^x('- 1 )= J2 x ^ V * " k= — oo

k= — oo

oo

oo

k= — oo

k= — oo

x«>= >T x<0cos(*«/T + p<0) «-*<'> = J T 4 ° e ^ r .

(3.75) v

'

Since it is regarded that x^'"1) is a given quantity, as we substitute it, in the capacity of an argument, in matrix (3.73), we linearize the latter. Now matrix (3.73) can be regarded as directly dependent on time. oo

A(x<'- 1 ))-A('- 1 )(r)= J2 2A<'-1)cos[fcI/r + V>(t~1)]-

(3-76)

fc=-oo

In this case the periodic matrix function A is represented in a harmonic Fourier series. Using (3.75) and (3.76) we rewrite equation (3.74) oo

(;M-Ar')40- E M-M^k n= — oo

k = 0, ±1, ± 2 , . . . , ±oo, for negative k h = 0. In this case I is a unit matrix of the second order.

(3.77)

Nonlinear resonance in radiophysical systems

179

A compact presentation of the latter system can take a matrix form: -L-C-l) •

A

-

x = f.

(3.78)

The bar above the symbols means that the infinite matrices and column vectors are of a block type, i.e. that their elements consist, respectively, of matrixes and column vectors of the second order. The matrix of system (3.78) is of the following type:

-]vl-ktl)

-A^1}

~-A(_j-1}

-A|'-x>

-A<'-1}

-A^l)

-At'- 1 )

-A*'' 1 *

(3.79)

juI-At"

The frequency multipliers of the first terms in the main diagonal of the matrix are of the type jkv, k = 0, ±1, ± 2 , . . . , ±oo. The infinite algebraic system (3.79) can be solved approximately by using the method of reduction. A system of three equations can be solved in the first approximation, five equations in the second one, etc. Continuing this process we obtain the g-th approximation as a result of solving a system of 2q + 1 equations,

4° = E 4° *'•""• k=-q

(3-8°)

The approximate solution of the initial equation is obtained from (3.80):

s<° = Rex(ol) = J2 4 ° c°
(3-81)

k=-q

The l-th. approximate solution can be estimated in terms of the scalar square of the solutions in the /-th and (/ — l)-th approximations:

A!=(x«-«r I I < o -« ( i - i ) ) = I f l*P + 4'-1'2 - 2 : « - » coe^W -rf-1 ')]• k=-q

The inequality A;
180

Nonlinear and parametric phenomena: theory and applications

If one knows nothing about the solution of Eq.(3.72), he should start with the zero approximation. It should be assumed that in (3.72) k = 0. For this quantity system (3.72) is integrated without difficulty. Then, following the procedure described above, the first approximation is obtained, then the second one, and so on until the l-th. approximation is obtained with the required precision. It is worth noting that the problems for obtaining all approximations of the solution of system (3.74), except the zero one, are of one and the same type. Coupled with numerical (computation) methods, the proposed approach can ensure a virtually arbitrary preset precision of the solution. The analytical approach proposed allows examining systems of nonlinear differential equations of an arbitrary type. We shall apply the analytical approach exposed above to the investigation of a serial resonance circuit with nonlinear capacitance C =

—— excited by sine 1 -(- Xq voltage U = U cos Lot. This is a typical problem, which appears whenever one uses semiconductor parametric diodes (varactors, varicaps), ferroelectric variconds, etc. as well as in the air capacitor cases, when a powerful source of electric energy is used for excitation purposes. Enormous forces undergoing periodic time-dependent changes with a double charge-altering frequency are applied to the capacitor plates. Such a phenomenon occurs, for example, at the output stages of the powerful radio transmitters for special purposes. The oscillating circuit equation concerning the capacitor charge is ^

(3.82)

+ 2a^+Lo20(l + \q2)q=jcosuJt,

where a is a damping decrement, LOO = y/LCo

, A is a small number, Xq2
Equation (3.82) is a particular case of Eq.(1.27) under section 1.4, if a = 0, A= UL-1, n = to. In our case the approximation we have used presupposes that the coefficient before q changes at frequency 2w, i.e. Q, = 2w. This yields the spectral composition of the charge in the form w + Ml = io{2k + 1),

fc=0,±l,±2,...,±oo,

i.e. in a stationary regime the charge can contain only odd harmonics of the frequency to. The solution of Eq.(3.82) and its transform are given in the form oo

q=

Y,

oo

Q2k+lCOs[(2k + l)L0t +
k= — co

£) k= — co

Q2k+i = Q 2 *+ie^ 2 f e + 1 ,

k = 0, ± 1 , . . . , ±oo.

Q2k+iej(2k+1),

Nonlinear resonance in radiophysical systems

181

The oscillating circuit equation concerning the charge transform is

g

+ 2 t ,J + u S ( 1 + A i±i),=

^.,

(3.83,

where q is the time function that is complex-conjugate with respect to q. For A = 0 the result is a zero approximation of Eq.(3.83): ^

« ^

+ 2

+

^

= ^»',

(3.84)

The precise solution of (3.84) is £(0) _

Q(0)e]Ljt^

Q(°>=

,

= - a r c t g ^ - .

U

As we know the zero approximation of the solution, we obtain the first approximation of Eq.(3.84):

+ *Q(°)eJ2«A q(D =

(3.85)

V-ei^

(the sign „*" denotes the complex-conjugate quantity). The precise solution of this equation can be presented in the form of an infinite sum. We shall limit our presentation to only four terms, which allow us to estimate the first and the third harmonic: » 1

( 1 )

-VO,n,e'

( 2 W ) u i

— /—I V2A:+l e it=-2

O (1)

e jV "+i

-O{1)

i V2fc+1 — V2ifc+le

T

•

As (f^> is substituted in (3.85) and the terms with equal frequencies are selected, a system of four equations is obtained, which is conveniently represented in the following matrix form.

|~0~

a-3-3 I a-3-i I 0 I 0 1 1 Q(±l I a-i-3

0

a-i-i

a-n

0

Q-i

I 0 I a 3 i [ q 33 1 [ Q^ I

_

_^_

[o

(»

a*\

182

Nonlinear and parametric phenomena: theory and applications

where a _ 3 - 3 = - 9 w 2 - w02 (l + | < 5 ( 0 ) 2 ) A W 2 *«»

Au,2.(0) 1

a-3-1 = —7-Q

-j6au,

a-i-3

=

-~A~Q

'

a . , - ! = -a; 2 + «02 ( l + ^ Q ( o ^ ) - j2a W ,

Aw2"'0) a-11 = ~T~V

Aa,g,(0) 1

ai-i

~A~^

=

>

a n = -<^ 2 + w02 U + ^<5 (0)2 J + J2aw, a 13 = ^ - Q , a31 = ^ Q ( 0 ) , ass = -9a, 2 +u,02 ( l + ^ Q ( 0 ) 2 ) + j6aa,. In order to solve system of Eqs. (3.86), we obtain the general D and particular determinants: D = a_3_ 3 a33(a_i_ian — o_uOi_i) — a_i_3a_ 3 _i(aiia 3 3 — 013031) —0-3-30-11013131, D-3 = f7L~1o_3_ia_na33, D-i

= —[/X~ 1 a_ 3 _ia_iia 3 3,

Di = f/L~1a33(a_3_3a_i_i - a_3_ia_!_3), D3 = -UL^1 a 3 i(a_ 3 _ 3 a_i_i - a_3_ia_!_ 3 ).

The complex amplitudes of the solution are expressed as Q-3 - - Q - . v - i -

£ , Vi - £ . y 3 - £ •

The transform of the first and third harmonics of the solution are obtained by means of the complex amplitudes:

^

= <# V** + Q%-3«\

f^ = Q^e'3ut +

Q-3e-^.

As the transforms are known, the harmonics themselves can be put down

gW = R4? = ReKQ^ + Q^y"1} = Qi1' cos(a;i + ^ 11) ), f3(l) = Ret] = ReKQ^ + Q^)^ 3 -'] = Q^ cos(3ut + ^ 31) ), where

Oi1' = y/tf?

coe„<» + 0 ^ COS/.W + (Q?rin^- Q ^ sm^j') 2 ,

tf," = ^ ( Q W COS VW + Q(i] c o s ^ ) ' + (Q^} sin^ 1 ' - Q™ s i n ^ ,

Nonlinear resonance in radiophysical systems

183

_(1) _ Q ^ s i n ^ - Q ^ W i ] *Vl "gWcos^ + Q^cos^r _(1) _ Qi"rin y ;"-qWri.^ **' C^cos^ + Q ^ W . T In the case of small values of A in (3.82) this can be the end of the analysis. Otherwise a second approximation equation is drafted with respect to the transforms

+£

(V + ^ ^ ) e 3 2 u t + V + ^ *» ) e ' 3 2 u t

^(1)^.(1) +

^(1)^(1)

Qi Qz e,-4u,« + Qi Q3 Z

Z

1\ e-j4Mt

( -(2) = ^ e J W t I

(387)

lj

The approximate solution of this equation is obtained by using the methods presented above. Approximations of a higher order can be obtained in an analogous manner. It is notable that the coefficients of Eq.(3.87) and the following approximations do not change the sign and that they are perfectly smooth. It follows from the theory of Fourier series that in this case the amplitude value of the harmonics desreases as the order increases, i.e. the sum of the Fourier series displays absolute and uniform convergence. In such cases it is not necessary to provide special evidence on the convergence in the solution of the infinite system of equations.

3.4. Implementation of parametric one-ports Parametric one-ports with a high-frequency change in the parameter are implemented mostly through the action of voltage or high-intensity current on nonlinear reactive and resistive elements and devices [78, 109]. Parametric oneports are obtained mainly by using nonlinear capacitive and active conductance of a p-n junction of a semiconductor (parametric) diode. Varicond and reactance valve capacitances, nonlinear inductances of ferrite cores and thin magnetic films, tunnel diode, Gunn diode, avalanche transit time diode conductances and transistor collector junction [78, 85, 235] are used less frequently. Surface semiconductor capacitances, ferroelectric structures, impedance converters with modulation in the

184

Nonlinear and parametric phenomena: theory and applications

reactive feed-back circuit, Josephson superconductor contacts, special capacitors with a light-beam controller capacitance, etc. [109] can also be used as parametric reactances. The major difference between reactive and active elements consists in that oscillation energy in an electric and magnetic form is accumulated and exchanged in reactive elements; at the same time resistances (positive or negative) are elements, which serve as a means of dispersing or increasing the energy in the system. The reactive and active one-ports with time-dependent parameters (parametric oneports) have both common and distinctive features. For example, the ability to generate harmonics is proper both to nonlinear and parametric capacitances and inductances, and to nonlinear and parametric resistances. A nonlinear element is also required, when alternating current is converted into direct current, but in this case the nonlinear element can only be a nonlinear resistance. A generalization of the basic properties of nonlinear reactive elements is provided by the theorem formulated by M. D. Karasev [236]: ,,It is not possible to obtain direct current from alternating current with the aid of a purely reactive element, even if this element is nonlinear. It is not possible to convert direct current into periodic alternating current by using a system consisting of an arbitrary combination of nonlinear reactive elements and constant resistances." A direct consequence of these principles is the postulate that it is impossible to design a negative reactance, which, having only its internal feedbacks and devoid of external sources of alternating electromotive force could impart energy to an external circuit. Negative resistance can be obtained artificially at the expense of a direct current source and a feedback controlling the conductance only under the condition that the total energy, which can be communicated by the negative resistance sector, exceeds the energy needed to control the feedback. The nonlinear capacitance of the p — n junction has become particularly common as a nonlinear reactive element in semiconductor parametric systems owing to its simplicity of design, compactness, low natural losses and the actual inertia-free barrier capacitance up to 1012 Hz [85]. It is a well-known fact already used in section 3.2, that the barrier capacitance complies with the following law of external voltage alteration [168]: C(0)

dq V

(3.88)

9k)

where C(0) is the differential capacitance at external voltage equal to zero, 7 = 1 , or — is a coefficient depending on the type of the p — n junction, and
Nonlinear resonance in radiophysical systems

185

which the current flowing through the diode is virtually zero. In the case of silicon diodes the values of Uf can fall in the range from 0.2 to 0.5 V, while in the case of gallium arsenide diodes Uf has higher values. From the viewpoint of the realization of parametric one-ports, the voltage Uf can be regarded as one of the parameters of the semiconductor diode.

Fig. 3.27. Volt-farad and volt-ampere characteristics of a semiconductor diode of the D817A type The parameter Uf, together with the parameter Ef, (the inverse voltage of p — n junction break-down) determine the possible maximum amplitude of the voltage Um at the diode in the absence of direct currents. We shall term the relation ay = coefficient of utilization of the (fk - Eo

characteristic of the semiconductor diode. The maximum value of the coefficient ay for a given negative bias Eg in the absence of direct currents is: a Vm

_Uf~E°
The coefficient avm grows with the increase of the negative bias to the maximum possible value (m)

_

Uf - Eb

Zifik - Uf - Eb

Fig.3.27 shows a regime of pumping with harmonic voltage (HVP). The other extreme case is harmonic current pumping (HCP). This regime requires a dependence of the diode barrier capacitance on the charge applied. As (3.88) is integrated at initial conditions U = 0 and 5 = 0, the following expression is obtained for the volt-coulomb characteristics of the parametric diode

9=Q*[l-(l-^)17],

(3.89)

186

Nonlinear and parametric phenomena: theory and applications

, _ C(0)y t where Qk = — ^ — . 1-7 As U in (3.89) is expressed through q and substituted in (3.88), the required relationship is obtained: —1

C(q) = C(0) (l - J - )

^

.

(3.90)

Fig.3.28 provides examples of a coulomb-farad and volt-coulomb characteristic of a semiconductor diode corresponding to (3.89) and (3.90).

Fig. 3.28. Coulomb-farad and volt-coulomb characteristic of a semiconductor diode of the D817A type There is a peculiar aspect that is worth mentioning. It follows from the voltcoulomb characteristic shown in Pig.3.28 that additional constant voltage should appear as a result of the nonlinearity of this characteristic. But in accordance with M.D. Karasev's general theorem [236], it is impossible to achieve detection (rectification) by using a nonlinear reactive (in this case capacitive) element. This apparent paradox is resolved by assuming that the high-frequency pumping results in the occurrence of an additional constant charge on the nonlinear capacitor that meets the requirement of zero additional constant voltage. The diode retains only the external constant bias EQ , while the constant charge at its capacitance changes

Nonlinear resonance in radiophysical systems

187

to Qo + AQo, where AQo is the additional charge caused by the action of the high frequency pumping. An analogy can be drawn with the HVP regime, when identifying the maximum value of the factor characterizing the utilization of the parametric diode characteristic in the HCP regime, i.e. (3.91)

Vfc - Vo

It follows from the expressions for a y m and agm, that at Uf =
(3.92)

Analogously for the opposite (inverse) capacitance (that is elastance) S(t) =

--^—(l-avcoSu;pt)\

Similarly, for the HCP mode we set q = —QE0 + Qm coso>pi in (3.90) to obtain C(t) = S{t) =

C(QEJ(1

- aq

cosujpt)^

x ^_ . -——{\-aqcosu}pty-i

(3.93)

188

Nonlinear and parametric phenomena: theory and applications

Regardless of the power supply mode of the diode, its capacitance and elastance can be presented through the following Fourier series (See also (1.25)): oo

C(t) = Co + 2 ^

Cn cos nupt (3.94)

"^ S(t) — So + 2 y Sn cos nu)pt.

The coefficients of the series (for example in (3.94)) will differ at ay = ctq, as follows from (3.92) and (3.93), hence the energy transformation in the devices employing nonlinear reactances will depend on the power-supply mode of the diode. Some studies show that at high pumping levels the HVP mode ensures more effective energy conversion in the broad-band modulation-parametric converters. In a mode of a partially forward-biased p — n junction, the sum total of the diffusion current and the current of the barrier capacitance flows through the diode. The diffusion theory of the flat p — n junction developed by Shokly is mostly applicable to lab germanium diodes. The distinctions from this model can be explained by the generation-recombination processes in the space charge layer, which can be actually accounted for by introducing a coefficient m in the analytical expression of the p — n junction volt-ampere characteristic [78, 231]: / qu \ I = IS f e^T - l j ,

(3.95)

where Is is the saturation current, U is the external voltage applied to the p — n junction, q is the electron charge, k is Boltzmann's constant, T is the absolute temperature. In addition, the components of the parallel equivalent circuit of the p — n junction are expressed in the following way: — differential conductance g (J

G

+ Js)yVl+a;27-' + l

= ~^kf

^

(3 - 96)

;

— diffusion capacitance Cd = q(* + Is">

-

•

mkT vVv'l + ^ + l '

(3 97)

— barrier capacitance

Ce = - r ^ T7 =Cofl-^lnff-l)l" 7> V

fkj

(3.98)

Nonlinear resonance in radiophysical systems

189

where r is the life-time of the minority carriers in the diode base, u is the circular frequency of the small alternating signal. Figs.3.29 and 3.30 present, in a double logarithmic scale [78, 231], some typical experimental dependencies of a p — n junction conductance and capacitance for a silicon and geranium diode in the current range of 1 - 100 fiA and a frequency range of 5 - 500 kHz. The solid lines show the theoretical dependence of the conductance G and the sum of capacitances C = Cd + Ce. For the sake of convenience during the comparison, there are additional entries along the x-axis: the direct voltage for the qU silicon diodes and the direct diffusion current Idif = IsemkT for the germanium diodes.

Fig. 3.29. Experimental (with different dots) and theoretical (continuous lines) dependencies of the conductance (a) and capacitance (b) of a silicic semiconductor diode of the D208 type: 1 - 500, 2 - 200, 3 - 100, 4 - 50, 5 - 15, and 6 - 5 kHz; m = 1,39, T = 14 //s

Fig.3.31 provides an answer to the logical question of how the dependencies of the capacitance C and conductance G of the p - n junction are going to change, given a further increase in the direct current. Fig.3.31 provides a clear picture of the values of the inverse currents /;, for which the capacitive nature of the differential reactance is changed to inductive. The dotted line shows the value of the ,,negative" capacitance obtained experimentally, while the dash-dot line indicates the equivalent inductance of the diode at frequency 500 kHz. kT When working with high pumping voltage amplitudes (3> ) and periodic

190

Nonlinear and parametric phenomena: theory and applications

Fig. 3.30. Experimental (with different dots) and theoretical (continuous lines) dependencies of the conductance (a) and capacitance (b) of a germanium semiconductor diode of the D7J type (m = 1,45 r = 14 /is); 1 - 500, 2 - 200, 3 - 100, 4 - 50, 5 - 15, and 6 - 5 kHz forward-conducting of the p — n junction, one should bear in mind the following typical aspects: 1. In the case of low frequencies

I w < — I, the small signal differential

V rJ characteristics G(/,w) and Crf(/,w) can be used, e.g. those shown in Fig.3.29 and 3.30, and as the frequency dependencies are taken into account a spectral analysis of the pulsative complex one-port can be made.

2. In the case of higher frequencies ( w >• — I, the effect of accumulating quasistationary charge of minority carriers in the diode base should be taken into account. Indeed, within the action time of the positive half-wave of the current, a charge of minority carriers is accumulated in the diode base. It fails to attain T . complete recombination over that time - t = — (T is the period of the highT frequency pumping voltage). Therefore, at time t = — the diode is in imbalance, i.e. it is a source of voltage AU(t), which causes the emergence of an additional current component, whose value is not set by an external source - it depends on the diode parameters, on the circuit, operation mode and frequency of the signal. The presence of a residual charge in the base is equivalent to a reduction of the

Nonlinear resonance in radiophysical systems

191

Fig. 3.31. Experimental dependencies of the differential capacitance (a) and conductance (b) of a silicic diode of the D208 type in the case of high injection levels: 1 - 500, 2 - 200, 3 - 100, 4 - 50, 5 - 15, and 6 - 5 kHz equilibrium potential difference of the p — n junction by AU(t). This results in a change in the constant current component both in value and in time, as well as in a bias shift. The influence of the effect of accumulation and establishment of a quasistationary charge of minority carriers on the performance of the semiconductor diodes in radio physical systems will be considered in detail in the following section. When the semiconductor diode operates in a direct current mode, its nonlinear differential conductance, rendered pulsative under the action of the pumping voltage, can be presented in terms of the small signal theory through a Fourier series (See also 1.25): 00

Gd = G0 + 2^G n cosnwt,

(3.99)

n=l

where

1 r

Gn — — / Gd cos nuitd(ujt). If the high-frequency voltage at the diode is relatively high, for example several volts, then in practice it is often sufficient to present the volt-ampere characteristic in the following simplified way: (3.100)

192

Nonlinear and parametric phenomena: theory and applications

The operation mode of the nonlinear conductance can also be convincingly described by using the concept ,,line segment angle":

n

U°

<3om = arccos — , where UQ is the direct voltage at the load connected in the circuit of the nonlinear conductance, Um is the amplitude of the alternating pumping voltage. In this case, the convenience stems from the fact that the line segment angle is unambiguously set by the ratio of the load resistance at the signal frequency (RT) and the internal (natural) resistance of the diode (Ri = —) through the formula: Gi tgQom-Qom _ Ri__ Yj_ 7r RT Gi

The real values of the ratio -=^— 0, 2 > —— > 0,001 encountered in practice RT

RT

have the following corresponding range of values for the line segment angle: 45° > e o m > i5°. As regards the parametric diode with external bias, the line segment angle is related to the parameter ay in the following way: cos0 o m =

-,

TpZ\-

a* i1 + i)

The concept of ,,line segment angle" can also be used when the parametric diode operates in an automatic bias mode. In this case, in order to determine the line segment angle, one has to measure the direct current Jo, which can be related to the amplitude of the alternating pumping voltage Um and 0 o m in the following way: h = —^-(sin0 r a - 0 o m c o s 0 ) , hence

j3irRjI0 Worn - y TJ V

um

The Fourier coefficients in (3.99) can be expressed through the angle of the line segment 0 o m in the following manner: G G C Go = T- L 20 om , Gi = - 1 2 s i n 0 o m , G2 = — L sin0 o m , 2TT

etc.

Z7T

Z7T

Nonlinear resonance in radiophysical systems

193

For the purpose of ensuring a high Q-factor in the resonance radiophysical systems, one usually uses a mode of inconsiderable transition into the direct current range of the diode characteristic. In this case the approximation of the volt-ampere characteristic (3.100) is quite crude, so the analytical expression (3.95) will be used. In this case the Fourier coefficients in (3.99) can be expressed as follows: Go = cJse-aE° J0(aUm) = alo, tri = -alse 2
°Ji(aUm) = -al0 2 Jo{aUm) atf">J2(aUm)

= -alo 2

T}

[, Jo(aUm)

etc., where Jn(x) are the modified Bassel functions of the ra-th order. Studies of Josephson's quantum-mechanical effect conducted in recent years have shown that the Josephson element has nonlinear inductance, which can be considered as a parametric element [140]. As first shown by Josephson, the phase difference if of the wave functions describing superconductive electrodes is related to the junction voltage V through the relation -y- = -^-V, 4>Q - ~— = 2.0710"15 Vb, qe is the electron charge, h is Planck's at

cpo

2qe

constant. The junction current may have a ,,superconductive" component (supercurrent) I3, which depends on the phase
L = \jnt,

(f>0

where L has the dimensions of an inductance: — = — cos<^, Lc = -——. L

Lc

2TT1C

With respect to small external signals the occurring supercurrent with its properties is equivalent to a reactive element with differential inductance L. The inductances that we are aware of are divided into geometric and kinetic according to the type of the energy stored. The geometric (magnetic) inductances are characterized by the capacity of current carriers to accumulate magnetic energy. Kinetic inductance has to do with the inertia of the current carriers. In reality, kinetic inductance does not occur in ordinary metals as a consequence of a relatively large dispersion of the current carriers.

194

Nonlinear and parametric phenomena: theory and applications

In the case of the Josephson effect, the inductance of the contacts is of a kinetic nature. Its nonlinearity is associated with a quantum-mechanical interferential effect of the wave functions. The typical dependence of the Josephson contact inductance on the phase ip determines the two basic modes of its utilization in parametric systems: with external pumping and with self-pumping. A major peculiarity of the Josephson supercurrent as a parametric element is the possibility to achieve considerable depth of parametric modulation. The Josephson inductance can take negative values for certain phases. On this basis a generic presentation of the effect of single-frequency non-degenerate parametric regeneration has been attempted [189]. The occurrence of this effect is associated with the possibility for the reactive parametric element to take negative values periodically in the process of its alteration. Section 2.1 has shown that similar phenomena are observed in modulation-parametric systems, where strong regeneration effects are manifested as a result of the reversibility of the interactions. Given a greater modulation depth, as in the case of a Josephson inductance with periodic transition to the area of negative values, there is no further need for a resonance circuit and the regenerative effects are also manifested in an aperiodic system. 3.5. Influence of the effect of accumulation of a minority carrier charge on the performance of semiconductor diodes in radiophysical systems The classical p — n junction diffusion theory developed by Shokli [168] assumes that the p — n junction has no resistive losses. Moreover, as diffusive conductance is calculated, it is regarded that the p — n junction is connected to a voltage source with zero internal resistance. Given this formulation of the problem, the charge of minority carriers in the diode base is ,,obliged" to follow the voltage applied, since the recharge of the diffusive capacitance conditioned by this charge should take place ,,instantly" through the zero resistance of the source. Consequently, this charge is always equal to zero at the beginning of each subsequent period of high frequency voltage and it is not accumulated at the diode base. Since the real diode always has finite resistance of losses, the mechanism of signal detecting and frequency dependence of the diffusive conductance of the p-n junction should be determined by the accumulation process. In addition, the wellknown definitions of ,,quadratic" and ,,linear" signal detection become meaningless. Usually, when the inertia properties of the semiconductor diodes are analyzed in the written sources, the transient characteristics, for which a voltage impulse is taken as an initial signal, are considered. But in the case of such radiophysical devices as detectors, rectifiers, frequency multipliers, parametric systems with semiconductor diodes in a direct current mode, etc., a quasi-harmonic signal is fed at the input and the issue of the choice of a diode cannot be unequivocally resolved on the basis of its transient characteristic, for in this case there is no unambiguous correspondence between the transient and the frequency characteristic.

Nonlinear resonance in radiophysical systems

195

Both theory and practice indicate that the semiconductor diodes in the resonance circuits of radiophysical devices can operate within quite a broad frequency range, and that they are also used at frequencies for which WT3>1, where r is the life-time of the minority carriers at the diode base. This possibility is mostly conditioned by the actual lack of inertia of the barrier capacitance of the p — n junction, up to frequencies of 1012 Hz, and the ,,top" limit is necessitated by the resistance of the semiconductor volume Rs. As the semiconductor diode operates in a transfer mode, for a part of the period the inertia of the diffusive capacitance and active conductance influence tangibly the frequency dependence of the detected voltage in the direct current area. In this connection, there arises a number of questions concerning, first, the legitimacy of using small signal differential parameters of the impedance and quasi-static voltampere characteristic of the p-n junction, when carrying out the computations in the kT conditions of high frequencies and considerable amplitudes of the voltage (^> — , q q is the charge of the electron), and, second, the peculiarities of using partially forward-biased p — n junctions in resonance circuits of radiophysical devices: the influence of the accumulation of a minority carrier charge on the nonlinear resonance in the oscillating circuit with a p — n junction in a direct current mode and on the exchange, in the parametric modulator, between the pumping circuit and the signal one in the situation of a parametric complex one-port, which shows inertia with respect to the pumping but is liable to modulation by the signal, etc. At this point we shall examine the process of accumulation and establishment of a quasi-stationary minority carrier charge at the diode base given a different nature of the load, the peculiarities of the diffusive detection mechanism at high kT 1 amplitudes of the alternating voltage (^> ) and higher frequencies (a; 3> —), and the influence of the effect on the diffusive conductance of the semiconductor diode. 3.5.1. Phenomenological model of the process of charge accumulation As we build a phenomenological model by using the charge method [78], we shall use as a basis the equations describing the behaviour of the carriers in the base area of the diode and providing a mathematically adequate description of the accumulation process. We shall carry out the calculations for the well-known model of a semiconductor diode with a plane p — n junction, subject to certain limiting conditions [168, 237]. Fig.3.32 shows the form of the current and the voltage at a semiconductor diode operating in a mode with an active load at high frequencies (w 3> —). We shall divide the task into two parts: the first one - from the beginning of the sine up to point £,r (Fig.3.32), as long as the current has a sine form, and the second one - from point £„• up to the beginning of the next positive period.

196

Nonlinear and parametric phenomena: theory and applications

Fig. 3.32. Form of the current (/) and of the voltage (U) on a semiconductor diode operating in a regime with an active load at high frequency (UJT = 50) Taking into account the familiar limiting conditions [168, 237], we can write an equation concerning the diffusion of the holes, the boundary and initial conditions in the following form: (3.101)

P(x,0) = 0,

P(oo,T c ) = 0,

^ ( 0 , T c ) = -sinfiT c at ^ OX

< T c < T

Si

^

+

^

;

il

P(0,Tc) = 0 a t ^ + T ; r < T c < ^ ± l ) , n = 0,l,2,... In Eq.(3.101) the concentration of the holes P is normalized to Pg = — , where j m is the amplitude of the current density, the time Tc and the coordinate x are taken in dimensionless units, Si = 2TTF, F is the frequency in units —. The boundary conditions are changed at moment TV, corresponding to £n (Fig.3.32). Equation (3.101) is solved by using a grid method and an implicit scheme, (Ax)2 with a step of Ax = 0,01 along the coordinate and a time step ATC = —-—, which differentiates the optimal speed of convergence. The calculation error has been established and it does not exceed 5%. Fig.3.33 reveals the dependencies of the charge concentration at the diode base at different points in the course of the first period. Curves 1, 2 and 3 correspond to different moments during the action of the positive half-wave. In the case of a negative half-wave the holes ,,collapse" back through the p — n junction, with a part of them recombinating. At the boundary of the p — n junction (x = 0 ) , the concentration falls to zero but a certain charge remains to be accumulated at the base in depth. Curve 6 illustrates the distribution of the concentration at the end of the first period. Fig.3.34 presents the dependencies of the residual charge at the diode base during the non-stationary accumulation process at the end of 1, 2, 3, 5, 8, 16

Nonlinear resonance in radiophysical systems

197

Fig. 3.33. Distribution of the concentration of the non-basic charge in the base of the diode at different moments in the course of the first period (o>r = 50) and 40 periods for frequency F = 8 (wr = 50). The process of accumulation and establishment of a quasi-stationary charge of minority carriers at the diode base is clearly seen.

Fig. 3.34. Distribution of the residual charge from non-basic carriers at the base of the diode in the case of non-stationary accumulation process (wr = 50) The dependence of the non-dimensional quasi-stationary charge QQ on the reduced frequency 9 = LOT is represented in Fig.3.35. The charge Qo reaches

198

Nonlinear and parametric phenomena: theory and applications

Fig. 3.35. Dependence of the non-dimensional quasi-stationary accumulated charge in the base of the diode on the reduced frequency maximum value at frequency 9 = 12. At a further increase of the frequency, Qo falls gradually and at u> —> oo it tends to zero asymptotically. Later on it will be shown that the charge method can serve as a basis for conducting a sufficiently thorough study of the frequency detection characteristic and the diffusive admittance of a semiconductor diode in an extreme mode - high kT 1 amplitudes ( > — ) and higher frequencies (w > —). The process of accumulation and establishment of a quasi-stationary charge at 27 the base of a semiconductor diode occurs at u> > —^— in the following way [237]. The charge left over at the base after the first period at zero initial condition 0(0) = 0 is Qi=Q*e~*-Qir, where QK is the unbalanced charge at time ut = n, Q\r is the charge that has run out during the negative half-wave of the first period, 9 = WT is the frequency reduced in accordance with the lifetime T of the minority carriers. After the second period, the charge left over at the base Q2 = Qi ( l + e - n r ) - ( Q 2 r - O i r ) . Analogous reasoning is applied to obtain the following recurrent formula for the n-th period:

_| or

Vnr

Qnr

^inr

J

_a

Qn = q ' e " T " a ? W - ( l - e - " » ) ,

(3-102)

Nonlinear resonance in radiophysical systems where QK =

m

199

(1 + e~~8 J , Im is the current amplitude.

It follows from (3.102) that the process of establishing quasi-stationary charge Qo at the diode base is determined by the lifetime of the minority carriers r according to the law (1 — t~r j . At n —> oo (3.102) yields

Qo =

Qne~f

-Qr 2^—\-e~S-

(3.103)

3.5.2,. Frequency characteristics of detection The rate of change in the detected current Jo, corresponding to the change in the frequency —— , is proportional to the charge Qo accumulated by the diffusive aw capacitance of the semiconductor diode [238, 239, 240]. In the general case dlo _dImK(6)

"fa -l^~7~

+

1

Imr

dK{9)

~

~M~ =2^°'

(3 - 104)

where K(6) = 1—

r is the frequency characteristic of detection, Qw = / I{t)dt Qw Jo is the charge of minority carriers injected into the base of the positive half-period, 1 [2T -*-O Qr{9) = — I I{r))e e dr\ is the part of this charge, which runs back through w Jo the p — n junction during the negative half-period, T is the period. If the diode operates with an active charge R, Im = —— and —-p- = 0. Then it dui

Q

0

=2/

r|^|.

m

(3.105)

In the case of a complex charge yL, the following can be written: ^

^ du

^

1

n

•K

dd

2TT

(3.106)

In the particular case of an automatic bias, the load represents a RC filter, and Eq.(3.106) takes the form K\d)

+V

JQ

~

orTT

= 7j—,

(3.107)

200

Nonlinear and parametric phenomena: theory and applications

where Qw = ^f-

= 2CUm, Um = ^ .

As (3.103) and (3.105) are equated and the denotation —— = ZR is introduced, Qw the following equation is obtained 1

dZR

6{e-e+e-o)

6(l-e-2-f)

d6

2(l + * ) ( l - H F ) "

R

Its solution is of the form

^R = —j— / —

where / ( ( ? ) = / '

^

7

^FV-^'

(3-108)

2

For the case of external forced bias E, the charge equation ( ——I

= I(t) )

yields: ^ 1-e T

QO=QVW-Q^

where <5T_2¥> =

7 m 7 -^ d2

/

I1+e

i +W V

7r-2yA

r

^ 1— e

« I cos

[

,

(3.109)

7r-2y"

e

^ 7

1 + e e~_

is t h e

charge at

moment wt = TT — 2y> for a zero initial condition, the angle

ZE = ^ ^ — / H{6)eF^de, 2 Jo where

F(6) = [cos

(3.110)

Nonlinear resonance in radiophysical systems

6)

/

7r+2y3

2TT\

e—9-+C-T

r

cos-0/?

i

H{6) =

1-e

j-1

201

7T—2^"

5~

^ ^

1.

For the case of an automatic bias mode the following is obtained for the relation between the direct and inverse charge ZRC(3.111)

where Vw "

i _)_ e

5—

The expressions (3.108), (3.110) and (3.111) are used to obtain analytical expressions for the frequency characteristic of detection and the quasi-stationary accumulated charge Qo for the different modes in the form

K{8) = 1 - Z(9), Qo{9) = - ^ ^ f ^ e - f - Z{9)\ . Fig.3.36 shows the theoretical characteristics of detection in a circuit with a semiconductor diode at different external negative biases. The dots indicate the experimental data.

Fig. 3.36. Theoretical frequency characteristics of detection with a semiconductor diode in an external negative bias regime. The experimental data are plotted by means of dots It is clear from the analysis shown that at high frequencies the detection mechanism is associated with the accumulation of additional quasi-stationary charge at the diode base. The emerging direct current /* = —- flowing along the active resistance of the load R, creates an automatic negative bias —Rig., which leads to the reduction of the potential difference of the p — n junction.

202

Nonlinear and parametric phenomena: theory and applications

Let us estimate the relation between the constant voltage u0, corresponding to Jg, and the amplitude of the variable component of the voltage Uj at the p — n junction itself. The direct voltage at the p — n junction is obtained from the voltampere characteristic:

- = M ^ + i ) ' -—*-£• <""> The amplitude of the variable component of the voltage at the p — n junction is Uj = Im\Zd\, where \Zd\ is the module of the diode resistance at high frequency,

,*,-[ w (*^ + 1 )vf. Then

^W^lnf Uj

IK

(3. 113 )

Is

V

'

The latter ratio shows that Uj does not depend on the amplitude of the alternating current Im and that it falls off as the frequency increases. At -f = 102, Q = 3 and KR(0) = 0,8, (3.113) yields ~ ~ 1,17, i.e. starting ls

Uj

at frequencies Q = 2, 7, the ratio — > 1 increases as the frequency goes up. 1' . It is obvious that at high frequencies Uj < UQ, and the diode is virtually always ,,forward-biased" for both half-waves (positive and negative) of the alternating signal. The analytical expressions for KRC{0) and KE(6), corresponding to the mode of automatic and forced external bias, can be checked through a numerical solution of the diffusive equation (3.101) for the adopted model of a p — n structure and for the conditions of a stationary regime. To this end, however, the overall transition process should be calculated many times, which, apart from the error accumulation, will require too much computer time. In order to avoid these inconveniences, we can work out an equation on the function of distribution of the minority carriers ip(x) at the diode base in a stationary regime. We present the solution at moment Tn (corresponding to moment £n in Fig.3.32) as a sum: 1) of the solution with initial zero condition and boundary conditions corresponding to the sine current h(x), where x is once again a coordinate determining the distance in depth of the diode base, and 2) a solution with homogeneous boundary conditions and an initial condition coming from the previous moment of switching H(x).

Nonlinear resonance in radiophysical systems

203

The expression for h(x) is obtained by solving the diffusive equation (3.101) at zero initial conditions by using an operator method on the basis of the CarsonLaplace transformations: k(x, T,) = _ _ /

V71" JO

sin[fi(Tw - Z2)]e

+

^ dZ,

(3.114)

where fl = 2nF, z is the impedance, which is external with respect to the diode. The following can be written for H(x) at the switching point T^:

H(x,Tn) = J™ G2(x,V) jjf%(OGi(«,£)#} dr,,

(3.115)

where ¥>(£) is the distribution of the minority carriers at the base and at the moment of previous switching, Gi(r/,£) and G 2 (z,'?) are Green's functions for the first and second boundary-valu problems [96, 100, 104]. The multiplier in the figure brackets represents the distribution of the holes at the end of the previous period determined on the basis of the certain distribution at moment T* of the same period. (3.115) is rewritten in the following way:

(3.116)

H(x,T,e)= n
where G(x, £) can be defined as Green's function for the problem with conditions switching at moment Tn. After certain transformations, the following is obtained

for
lyJitlQ

V.

\Z\/1-Kl2-Lo /

\Zy/llrl2-L0/

I J

where To is the voltage period in units r , T2 = TQ — T^. For the stationary process the solution obtained for the switching moment Tff is identical with the distribution
V(x) = h(x)+

f°°

v(QG{x,Z)dt.

(3.117)

The expression (3.117) is Fredholm's equation of the second order, which can be used to establish
204

Nonlinear and parametric phenomena: theory and applications

In order to obtain the solution in the case of an automatic bias mode, it is sufficient to simply transform the function h(x):

h{x,Tn) = ^=f V ^ Jo

7r{sm[n{Tn-z2)+
(3.118)

The current at an arbitrary moment after TK is determined according to the formula e-Tc

Q

_j2

/-oo

I(Tc) = -7rP(x,Tc)x=0=


(3.119)

T'c = Tc — I Tn + 27T— ), S = 1, 2, 3,. . . , Tc is the current time in units r. The harmonics of the current can be calculated by simply using the Fourier formulae. Knowing the charge at the base of the diode Qrn at moment Tn, we can resort to the charge equation ~+= I(Tc), dlc T to determine the charge at an arbitrary moment as Q(TC) = Qe-T'

(3-120)

+ e~T< f ° e'/(v)di7, Jo

where C; is determined by the equation Cl

=

QTweT*-

Jo

e^sini^*?-

For the purpose of calculating the charge harmonics, a relation can be established between them and the current harmonics. Multiplying the left-hand and right-hand parts of Eq.(3.120) by sinnft and cos nft successively, and averaging for a period, we obtain Qna~

_ Inc - nQ,Ins Ins + nillnc l + n2fi2 ' W»«=- i + n 2 ft 2 '

,,.„., y6AZl>

where Inc, Qnc are the cosine and /„„, Qns the sine components of the current and the charge, n = 1, 2, 3 , . . . is the number of the harmonic. Equation (3.117) has been investigated via numerical methods and its error does not exceed 5%.

Nonlinear resonance in radiophysical systems

205

Fig. 3.37. Distribution of the non-basic carriers in the base of the diode at the moment of switching TT in the case of a stationary process, for frequencies: 1 - F = 0, 03; 2 - 0,1; 3 - 0,3; 4 - 0,7; 5 - 1; 6 - 3; 7 - 8 and 8 - 12 Fig.3.37 shows the distribution of the minority carriers at the diode base at the moment of switching TV for a stationary process and for different frequencies. The charge reaches its maximum at frequency F = 0,3 ( 0 = LOT ~ 2). At F = 0,7 ( 0 ~ 4) the maximum concentration of the minority carriers is higher but the charge is smaller than in the case of F = 0,3. When the frequency increases, the charge decreases, and the switching current remains virtually unchanged. The dotted lines in Fig.3.37 indicate the distribution of the minority carriers at frequency F = 3 and 8 at the moment of switching during the first period. Fig.3.38 shows the calculated reverse current of the diode in a stationary regime for different frequencies. There is a reverse current impulse even at F = 0,01 ( 0 ~ 0, 06) but, as the Figure reveals, the accumulation of additional charge from period to period begins only at frequency F = 0,45 (0 ~ 2, 7). The experimental study is fully in concord with these conclusions [239]. It is worth noting that the theoretical frequency characteristics of detection, obtained on the basis of a numerical solution of Eq.(3.117) and by using the analytical expressions (3.103), (3.108), (3.109), and (3.110) (Fig.3.36) coincide with at least 5% precision. 3.5.3. Diffusive impedance of the p — n junction at high signal amplitudes and higher frequencies In order to determine the diffusive components of the impedance of the p — n junction in the situation of accumulated quasi-stationary charge, we shall express the amplitude Uj of the alternating voltage at the p — n junction in the following

206

Nonlinear and parametric phenomena: theory and applications

Fig. 3.38. Reverse current of the diode in stationary mode; / without bias, / / - in t h e case of external forced bias pE = 0,75, for frequencies: 1 - F = 0,03 ( 0 sa 0,2); 2 - 0,3 (2); 3 - 3 (20); 4 - 8 (50) way [241]:

JJ. 3

hn

=

[ff ( e)£ + j.]AV5

~

^

AV5tf(e)(go+j.r)'

(3 122)

where A = — , K(Q) and Qo are the frequency characteristic of detection and the quasi-stationary charge for the different modes considered above, Is is the reverse conductance current of the diode. It can be seen from (3.122) that the charge at the base of diode Qo is related to the voltage at the p - n junction by a relationship different from that known in the classical diffusion theory [168]. The diffusive capacitance of the p — n junction is determined as

CfP-n = ^T = I.TAK(Q)VQ— dUi Given UjV&K(e)-

-r*.

,r[i-tf,Vetf(e)£]

(3.123)

< 1, i.e., when if < 1, where 7® = — , the diffusive

capacitance of the p - n junction is Cfp_n = I s Ar

\/Q and it does not depend

on the amplitude of the alternating voltage Uje (for example at 0 = 50, Is — 10~5 A and r = 10~5 s we determine Cfp_n ~ 280 pF).

Nonlinear resonance in radiophysical systems When Im > L^QZ,

207

ie. / oe > / „


(3.124)

In this way, in the case of high signals, Cf n depends on Im as ~ J^, and it goes up quickly as the amplitude of the current increases. The frequency dependence of Cfp_n is not identical with the classical (Cd ~ -7=) and, as follows from (3.123) and (3.124), it is more complex in V0 character: in the case of small signals C® n ~ K(Q)y/Q = a&2 -\- 6 0 ~ I , and at higher signals Cfp_n ~ K3(e)Ve = (a0S + 60"f) . In an analogous way, the diffusive resistance of the p — n junction can be determined as R%-n = | ^ = 7

^ 2

•

(3-125)

If /m < t-KWyLe-tf « \then ^ - = ijjsObviously, at small signals, Rjp_n coincides exactly with the classical formula for differential resistance for an ideal p — n junction.

Given J m > ^ ^ y , ie- I® > L, R%-n = j 4 j 0 A / Q ' a n d t h e

f r e q. u e n c y

resistance of R®p_n is retained within the classical framework (Rj ~ -7=), while Ie the absolute value goes down —r- times, which is indicative of the existence of strong s modulation of the conductance at the expense of the accumulated charge. The dependencies obtained yield an expression for the diffusive time constant at high frequencies (Rdp-n^dp-n) =

r>

(3.126)

while the classical diffusion theory gives (RdCd) — —, which does not depend on r. When the p — n junction is used in an automatic bias mode as a capacitor in oscillating resonance systems, as the analysis shows, the diffusive capacitance Cf n largely determines the characteristic bifurcational peculiarities of the nonlinear resonance at high amplitudes and higher frequencies u ;§> —. T

Fig.3.39 shows the dependence of the effective capacitance of a semiconductor diode in an automatic bias mode (the experimental data are marked with different

208

Nonlinear and parametric phenomena: theory and applications

Fig. 3.39. Dependence of the effective capacitance of a semiconductor diode (D208) on the voltage amplitude at a high frequency (© = ur = 120) for the values of the resistance R of the iJC-filter automatic bias, as follows: 1 - 30 Mfi; 2 - 3 MQ; 3 - 0,33 Mfi dots). A comparison of the diagram in Fig.3.39 with the dependencies of Cef at low frequencies ( u < — I - for example Fig.3.6 - shows their qualitative similarity. V TJ

As shown in Chapter 1 and Chapter 2, when the purpose is to promote the effectiveness and expand the functional capacities, the parametric diodes operate in a partially forward-conducting regime with small direct currents. At first glance it

may seem that when the pumping frequency wp > —, the p—n junction cannot play the part of a complex parametric one-port, since the conductance (as well as the diffusive capacitance) fail to change in accordance with the pumping frequency. It is not difficult to show, however, that the modulation of the quasi-stationary charge with signal frequency (in the modulation-parametric systems the signal frequency is much smaller than the pumping frequency) provides a possibility for obtaining all functional regularities typical of the four-frequency parametric systems with inertia-free complex parametric elements (See Chapter 1 and Chapter 2). Thus, for example, the charge of the signal frequency can be defined as a variation of the constant component of the charge at the p-n junction. We have qi

= 6{QE + Qo) =

Wo

ft

+ —gff^

SUP,

(3.127)

where QE is the constant term in the Fourier expansion of the barrier capacitance charge, UQ is the constant voltage at the p — n junction, Um is the amplitude of the pumping voltage, 6UP is the variation of the pumping voltage, which is essentially the sum of the voltages of the combined frequencies.

Nonlinear resonance in radiophysical systems

209

The expression (3.127) can be rewritten as qi

= (CE + C°)U^ + (Ci + Cf)6Up,

(3.128)

where CE and C\ are the constant terms and the first coefficient in the Fourier expansion of the barrier capacitance, which is subjected to the action of the pumping voltage, C® = -T7— is an addition to the average capacitance conditioned by the Uo accumulated charge QQ, Cf=-jr—— is the diffusive capacitance of the semiconductor diode at high amplitudes of the pumping voltage and higher frequencies. In an analogous way, the diffusive current of the signal frequency can be presented through the variation of the constant components of the current

e

11 = 6I° =

dl*

MUl

dl?

+

duZ6Up'

or (3.129)

i = G°U1+G%6Up,

O dl® where G® = ——- is the average conductance of the p — n junction, G? = ° TUQ _ oUm is the diffusive conductance of the semiconductor diode at high amplitudes of the pumping voltage and at higher frequencies. Taking into account the active losses in the oscillating circuit of the pumping section expressed as series active resistance Rp, as well as the formula for the impedance module of the p — n junction \Zd\, the amplitude of the pumping voltage can be determined as

„

P., (£+'•)*>** + '

In the case of a high-frequency pumping corresponding to the condition wp > —, the conductance of the forward-biased p — n junction virtually fails to change with the pumping frequency and its harmonics, so the coefficients Gi and G2 in the matrix equation of the parametric modulator (1.26) can be regarded as equal to zero. Taking into account (3.127) and (3.129), the parametric modulator in this case can be described by the following matrix equation [231]:

j, 0 0

j/! =

i^+Ci -jw-Ci

Gf + jUl(d y+ -ju-C2

+ Cf) Gf + jWl(Ci + Cf) U, ju+C2 V-

U+ . U-

(3.130)

210

Nonlinear and parametric phenomena: theory and applications

If we conduct a study of the input impedance of the parametric modulator in a way analogous to that developed in 2.1 — and if we use (3.130), we obtain for the conductance (G®) and susceptance (-B®) introduced in the input circuit:

,n2re , 4Q2(mx + m ^ 2 + Md[l + Q2(*' - 4£2)] Gg = -AQ G rmio [1 + Q2{H,_4e)]2 + i m y

re

R0

, ,(™i + md)[l + Q V - 4£2)] - AMd

,nzrenl

where md =^,Md

= | | , Coe = CE + Co - C2, G@ = Gf +

m _ (3-13D

(3.132)

G0-G2,

*' = 4(e02 + tom2). The effective (equivalent) negative capacitance introduced into the signal circuit, will obviously be written as C9 = -AQ m^C0

[1 + Q2{H,_Ae)]2+m)2

,

(3-133)

In the bandwidth (x' ~^> 4^ m , ^ m is the upper boundary spectral component of the signal) (3.134) C9 = - 4 g m^oCo

(1 + Q2X)2

•

(3.135)

The maximum value G® and Cf realize at detuning £o = j§:

,

| a

r

e

c *- max

_

nrn

r&

"

QmiC°

(3.I36)

=9^Mi,

max

1 + Qmi

\mi+md

2Md

1

[ T T o ^ " (i + Qm2y\'

(3-137)

hence, at m2 ^ ^o G®

(m)

= QG@miMd,

(3.138)

max (m)

Cf

= g^miCm! +md- 2Md).

(3.139)

max

Dependencies G® and C® from (3.131) and (3.133) repeat qualitatively the respective dependencies (2.6) and (2.10) - Fig.2.5. This result shows that owing to the effect of accumulation and establishment of a quasi-stationary charge at the base of the parametric diode it is possible to use a parametric one-port, which is of an inertia type as regards pumping but liable to modulation as regards the signal, for modulation-parametric amplification.

CHAPTER 4.

CHAOTIC OSCILLATIONS IN RADIOPHYSICAL SYSTEMS

4.1. Radiophysical systems with natural complex dynamics. Research methods 4-1.1. Major concepts: bifurcations, chaos, strange attractor, fractal dimension. Conditions for the manifestation of chaotic behaviour in radiophysical systems The introduction of the concept ,,chaos" and its discovery and investigation in a broad class of dynamic systems, including those in Mechanics, Radiophysics and Electronics, has been one of the major achievements of the physics of nonlinear oscillations in recent years [3-31]. A dynamic system usually implies a physical object, a mathematical model or a process, whose condition is determined by the totality of some quantities at the initial moment; furthermore, the state of the system at an arbitrary point of time, set in advance, can be established with the help of a given functional operator. The chaotization of oscillations can be defined as the occurrence of peculiar non-homogeneous (irregular) motions in absolutely determined oscillating systems. The describing equations have limited non-periodic solutions of a random nature even though there are no random parameters in these equations. Apart from the heuristic nature of the problems related to the investigation of chaotic oscillations, the latter is important for the engineering and research practice due to a number of other reasons as well. First, the chaos or noise in the radiophysical systems impedes the prediction of the length of their operation and their reliability, since the exact dependencies of the voltages and currents on time prove to be unknown. Second, realizing that the ,,ordinary" nonlinearities can lead to the development of chaotic modes, we are faced with issues related to the value and authenticity of the numerical modeling of nonlinear radiophysical systems in general. It is usually believed that the more powerful supercomputers are used, the more precisely the behaviour of the system can be predicted. In nonlinear problems involving chaotization, however, the behaviour of the system is highly sensitive to the initial conditions and the precise calculation of the future behaviour may prove impossible even in the case of periodic modes. Mechanics, Radiophysics and Electronics are those areas of human knowledge where the phenomena of complex dynamics are most widespread. At the same time, these scientific areas provide the best possibilities for complex analytical, numerical and experimental research. Besides their general scientific and world-outlook value, the investigations of the chaotic oscillations in the mechanical, radiophysical and electronic systems specifically are of great practical significance in the analysis 211

212

Nonlinear and parametric phenomena: theory and applications

of regime stability and reliability in generators, amplifiers, SHF and Josephson systems, bistable dynamic systems, in the development of sources of stochastic and noise signals, etc. While the possibility for chaotic behaviour in the absence of external fluctuational (random) action is regarded as more or less obvious for the systems with a large number of degrees of freedom, in the case of systems with a small number of degrees of freedom (for example with 1.5 degrees of freedom) this possibility has proved to be unexpected for the researchers. Hence, the most extensively studied issue in recent years has been the chaotic behaviour of systems with an ultimately small dimension of the phase space TV = 3 (both autonomous systems described by differential equations of the third order and non-autonomous systems with one degree of freedom, subjected to external determined periodic action). Until recently, it was believed that the behaviour of the simple systems of the type of a radiophysical nonlinear oscillating system or a generator of radiophysical signals is completely predictable for a given arbitrary time interval by virtue of the determinateness of the equations that describe them. It turns out that chaotic oscillations may occur in such systems without external random action as a result of complex natural behaviour of the system brought about by the instability of the phase trajectories. The chaotic oscillations can be qualified as non-homogeneous motions characterized by irregularity, unpredictability and randomness. A well-known fact from the Theory of nonlinear oscillations [44-48] is that the dynamics of oscillating systems is reflected through the totality of phase trajectories in the phase space determining the phase portrait of the system. Each attracting or repelling set in the phase space is called an attractor. Such attractors can be the stable and unstable peculiar points, limit cycles, separatrices and other sets. In the case of chaotic oscillations, certain phase trajectories prove to be unrealizable as a consequence of a peculiar type of instability but the totality of phase trajectories as a whole is realized in a definite limited area of the phase space. Since it is not simple for one to imagine all this, the geometrical image, consisting of such a totality of phase trajectories, has been termed a ,,strange attractor" a strange attracting set. Phase trajectories are internally unstable, the proximity between them, typical of determined systems, is not retained, most frequently they ,,diverge" exponentially. Yet, for an arbitrarily long interval of time the totality of phase trajectories ,,shrinks" in the direction of the set that attracts it - the strange attractor. A strange attractor can be defined as a geometrical object in the phase space, where all phase trajectories of the system are tending to, and which, at the same time, repels them as a result of their instability in the area of this geometrical object. Obviously it is not possible to realize simultaneously exponential instability and ,,return" of the phase trajectories typical of chaotic oscillations, on the phase plane, since the phase trajectories do not cross each other on this plane. That is why the minimum dimension N of the phase space at which chaotic oscillations

Chaotic oscillations in radiophysical systems

213

can exist is N = 3, which corresponds to an oscillating system with 1.5 degrees of freedom. Chaotic oscillations can also be excited in an ordinary passive nonlinear oscillating circuit with 1.0 degree of freedom, when the circuit is subjected to the action of an external determined periodic signal. In this case it is also regarded that the system has three-dimensional phase space, with time playing the part of a third coordinate. The behaviour of the systems with chaotic dynamics is sensitive to the precision of setting the initial conditions and this is indeed one of the reasons for its unpredictability, i.e. regardless of the fact that the description of the system is fully determined, it is not possible to predict its evolution in time. It is necessary to develop a special statistic theory of the dynamic systems with a strange attractor. A typical concept of the Theory of nonlinear oscillations is bifurcation denned as a change (most often leap-like) in the equilibrium state and stability of the system, or a change in the number and stability of the solutions of the describing equation. Bifurcation is a typical nonlinear phenomenon. In the classical Theory of probability, the existence of randomness and respective probabilities is postulated and the issue of their origin is not raised. The canonized axiomatics of the classical Theory of probability operates with the concept of stochasticism and it is accepted that the latter is engendered in open dynamic systems in a fluctuating medium. Until recently it was accepted that all phenomena in our surrounding world were divided into determined and random ones. The former were studied by the Theory of differential equations, classical Electrodynamics, Mechanics and Mathematical Physics, while the latter were considered by the Theory of probability, the Theory of random processes, Statistical Physics and Quantum Mechanics. Such a strict division seemed indisputable and raised no doubts. At present, views are radically changing. Moreover, the development of the Theory of nonlinear oscillations and the Theory of dynamic systems naturally leads to the issue of the origin of randomness and the essence of probability, and to the design of respective new methods for resolving the emerging problems. At present, stochastics in continuous dynamic systems is mostly associated with the impossibility to achieve an exact repetition of the same initial conditions and the considerable influence that can be exerted even by the smallest perturbation. In this sense we can maintain that the stochastic and chaotic motions of determined dynamic systems are the ,,real" randomness in our surrounding world. These new ideas are taking shape only due to the discovered possibility for stochastization and chaotization of the motions in determined dynamic systems. Obviously, this is a colossal set of problems, whose elaboration has just been initiated. Every day, the modern scientific world is flooded by a huge amount of new facts and events in this area. The mechanisms of the occurrence of stochastic behaviour in the simplest determined dynamic systems are subjected to particularly precise and comprehensive consideration, and new initial paradigms affecting our deepest intuitive notions are worked out. The term ,,chaotic", as used in modern publications, applies to those motions in determined mechanical, physical and

214

Nonlinear and parametric phenomena: theory and applications

mathematical systems, in which lack random or unpredictable forces or parameters, while at the same time their phase trajectories demonstrate great dependence on the initial conditions. In order to grasp more profoundly the meaning of the stochastic behaviour of oscillating systems, one should first answer the question in what way chaotic motion, i.e. a random process, may arise in systems without fluctuations and with a minimum number of freedom degrees, described by determined differential equations. It follows from the theorem of the existence and uniqueness of the solution, that in a set of strictly given initial conditions the differential equation has a definite determined solution. It is not possible to obtain random function after solving the equation. It turns out that the answer lies in the instability of all or almost all solutions. But even if all solutions of the equation are unstable, for each specific initial condition one invariably obtains a solution in the form of a determined function. Then how can the instability of the solutions cause stochasticism? The point is how one understands stochasticism. Currently the concept ,,determinism" is rationalized in the following aspect. It is believed that the state of the system cannot be specified with infinite accuracy, hence the initial conditions will contain certain indeterminacy e. It is clear that the solution of the differential equation will also contain indeterminacy (finite dispersion) depending on e and on the time t. If initially e is set as tending to zero and then the equation is solved, this will imply that it will be solved under initial conditions set rigorously, identified with infinite accuracy. In this case, the result is always utmost predictability. From all appearences, this track is not practicable and should be replaced with the following rule: first, given a set e, identify the behaviour of the trajectories and areas of finite dispersion for an arbitrary t, and then set e as tending to zero. In addition, the change in the final dispersion at t —* oo should be traced. If at e —• 0 the extreme dispersion tends to zero (and such will be the case whenever the solution is stable), the behaviour of the system will be determined. If, on the other hand, at e —> 0 the dispersion of the trajectories does not tend to zero for t —> 0 (and this is observed, if there are unstable solutions), the behaviour of the system will be ,,unpredictable" and it will be described by a random temporal function. This understanding of determinedness differs from the traditional one in that the order of the limit transitions e —> 0 and t —• 0 is changed. Since the evolution in chaotic systems is unpredictable, the question whether it is relevant even to talk about any dynamic structures stands out. The answer to the question on dissipative systems is affirmative: chaotic motions have a new geometric property called fractal structure. A definite internal order has been discovered in the chaotic dynamics. It conditions the possibility to predict, study, and, when necessary, suppress the chaotic noises in radiophysical and other systems. It is believed that just as dynamics, the new science of the 17th century, called to life differential calculus, nowadays nonlinear dynamics calls forth such new ideas from the areas of geometry and topology as fractals which should a 21th century

Chaotic oscillations in radiophysical systems

215

researcher fail to grasp them, he would remain unable to understand the subject of his own investigations comprehensively. Currently there are two most popular qualitative indicators of the existence of chaos: a positive exponential indicator of Lyapounov and a fractal structure of the orbit in the phase space. The peculiarities characteristic of the chaotic oscillations can be most readily observed by using the cross section of Poincare. It provides a picture of the cross section of the attractor in whose area the motion in the phase space is focused. When the motion is chaotic, a complex labyrinth-like multi-leaf structure occurs. The set of points delineating definite lines have a more subtle structure as well, which can be analyzed given a greater magnification of a certain local section. Such Poincare cross sections are said to have fractal structure. What is known as fractal dimension is used to characterize the strange attractors and measure the fractal boundaries in the areas of initial data and parameters. Currently, there are a number of fractal dimension definitions: capacitive (or geometric), mean score, correlational, informational, Lyapounov dimension. In all cases, the calculation of the fractal dimension is based on digitalization of the signals in equal time intervals. The fractal dimension is meaningful only with respect to dissipative systems. It is typical of them that orbits tend to fill only a part of the phase space. Fractal dimension is a measure of the extent to which, a certain space is filled by the orbit. In a more general perspective, the dimension determines the amount of information necessary for setting the coordinates of a point belonging to the attractor. In the case of classical (regular) attractors, the dimension is an integer: the dimension of a fixed point is zero, that of the limit cycle is unity, while that of the two-dimensional torus is two, etc. The existence of a fractional (non-integer) dimension is indicative of the presence of a strange attractor. The general cause for the occurrence of chaotic motion in dynamic systems is the loss of stability and the exponential ,,scattering" of the close phase trajectories combined with their general limitation and general shrinkage. The exponential increase in the deviation followed by shutting down can serve as the simplest model of instability coupled with general shrinkage. One can imagine that there is an area in the phase space G, where the phase trajectories diverge and as a result leave the area. When the phase trajectories leave the area G, the nature of the motion changes in such a way that the phase trajectories go back to the initial area G. This return to the area G is the very phase of ,,shutting down", when the deviations that have occurred are restored. This cycle is repeated many times. It is noteworthy that the controversy between these two tendencies - instability with a rapid process of increase in the oscillations and a limitation of that increase - is also typical of the strictly periodic oscillations (in resonance or generator systems). The causes for the instability and the increase in the oscillations may be the same in both cases of determined and chaotic oscillations as well. The difference lies in the nature of the limitation: in the first case, this is gradual slowing down and restriction of the oscillations, most frequently leading to the formation of a stable limit cycle in the phase space, while in the second case it is the ,,abrupt" shutting down.

216

Nonlinear and parametric phenomena: theory and applications

4-1-2. Basic mechanisms of transition from determined to chaotic oscillations As far back as the end of the 19th century, the great French researcher A. Poincare suggested [91] that randomness (indeterminacy) may occur as a consequence of the instability of the very motion in the system. In such an unstable system, ,,an absolutely trifling cause, escaping our notice due to its insignificance, can bring about actions that we cannot predict in advance". Chaotic oscillations were first discovered in 1963 in the Lorentz system, already regarded as classical. The latter describes the thermal convection in a horizontal liquid layer heated from below. Lorentz's work, whose significance is generally acknowledged today, was hardly known for many years after its publication (till 1975). Today the system of equations examined by Lorentz is called ,,the Lorentz model". Nowadays it has been shown that the system of Lorentz equations describes a number of radiophysical generator and converting systems, where analogous phenomena occur. Chaotic behaviour of systems, described by means of a strange attractor, has been encountered in numerous radiophysical and electronic systems, as well as in mechanics and biology, in economy and in the organization of society, in ecology, hydrodynamics and many other areas of science and technology. Several models of bifurcations [17, 19] are known for their ability to cause, for example, the emergence of chaotic oscillations in a radiophysical nonlinear resonance system. The first mechanism has been proposed by Reuel and Takens and it consists in a three-dimensional dynamic system (described in a three-dimensional phase space) that passes from a state of stable equilibrium through three successive bifurcations: a limit cycle - a two-dimensional torus - a three-dimensional torus. There is a considerable probability that a strange attractor (chaos) will occur afterwards. The second mechanism has been developed by Feigenbaum. When a certain bifurcation parameter, K, is changed, the limit cycle becomes unstable. A new, two-revolution stable cycle appears. Given a further change in K, the following bifurcation occurs: the two-revolution limit cycle loses its stability and a new fourrevolution cycle comes about. If parameter K is changed further, such a critical value, K = Kcrj is reached, beyond which an infinite number of unstable cycles with different periods emerge in a limited area of the phase space. Besides, it has been established that in spite of the specific peculiarities of the system and, in many cases, despite the dimension of the phase space, the value of the bifurcation parameter is governed by the Feigenbaum relation l\n - l\n-\

o

where n is the number of the bifurcation. The third mechanism can be called an intermittency mechanism. The stability of the limit cycle is lost as a consequence of its merger with the unstable one.

Chaotic oscillations in radiophysical systems

217

Periodic oscillations gradually disappear as a result of their interception by chaotic ones. A lasting laminar process, alternating with chaotic oscillations, is observed under this mechanism. The fourth mechanism is associated with firm occurrence of chaos around a stable singular point. Such a bifurcation is termed ,,reverse Hopf bifurcation". Its essence consists in that the unstable limit cycles, which occur around the stable singular point, penetrate into these points during the next bifurcation. It is in such a way, for instance, that the chaotic oscillations emerge in the classical Lorentz system. We can also imagine that a complex, non-homogeneous motion, can be caused to happen by resonance overlapping. If an external force, p(t), with a period -tracts on a linear oscillating system, and if it is described by x+u>lx=p(t),

(4.1)

where UJQ is the natural resonance frequency, then, as it is well known, the solution of the homogeneous equation is x0 = A cos(o;oi) + B sin(o>0t).

(4-2)

If the external periodic force is presented in a Fourier series, oo

p(t) = — + Jjfan cos(nflt) + bn sin(nfit), then, as we carry out a harmonic expansion of the particular solution of the describing equation (4.1) as well, we obtain for the nth. term

n =

an cos(n£lt) + bn sin(refti) ^ - n2ft2 '

Given WQ ~ n2Q,2, there are resonances causing the increase in the amplitudes of the respective spectral components. If the oscillating system is nonlinear, all harmonics, multiple to the base frequency, WQ, will be present in the homogeneous solution (4.2). In this case resonance effects appear whenever the ratio — is a rational number, thus a dense system of resonances emerges. Since the natural frequency of the oscillations in nonlinear systems is a function of their amplitude, while the resonance phenomena change this amplitude and, therefore, the frequency of the spectral components, a peculiar counteracting process of violating the conditions of resonance manifestation occurs. This intricate interdependence obviously changes the nature of the phase trajectories and causes a complex nonhomogeneous (irregular) motion in the system.

218

Nonlinear and parametric phenomena: theory and applications

Jf.1.3. Methods and criteria for identifying chaotic oscillations A number of methods are used for investigating the nature of the motion in specific oscillation systems for the purpose of establishing the presence or absence of a strange attractor. 1. Numerical integration of the differential equation describing the system except for the transient process. The result is the picture of the time-dependent change of one or several variables - x(t), x(t), etc. 2. Investigation of the Fourier spectrum of the oscillations, most frequently by using a quick Fourier transformation. It is possible to trace, for example, the evolution of the oscillations in a radiophysical system from periodic to chaotic ones, when changing one or several controlling parameters, such as amplitude and frequency of the external action, coefficient of nonlinearity, damping decrement (increasing increment), etc. Initially an increase in the level of the harmonics is observed in the spectrum, then subharmonic components appear (bifurcation of the period of oscillations). A continuous spectrum takes shape in the chaos, and only several characteristic frequencies of the system can be discerned. 3. Investigation of the correlation functions of the chaotic oscillations. Given a continuous spectrum, the correlation function for an arbitrary phase variable Xi(t) is of a declining nature by virtue of the Wiener-Hinchin theorem. The correlation function can be calculated directly from the formula k(r) = < XiXi+r - m2 > , where m is the average of the variable Xj (rn = of;). 4. One of the criteria for the existence of chaos is the positive sign of the entropy K of the dynamic system, which is characterized by the average speed of divergence of the phase trajectories in the strange attractor, _, .. .. M(e,t) K = lim lim —-—- , £—•0 t—»OO

t

where e is the inaccuracy of the measurements, t is the time of observation, M(e, t) is the maximum number of trajectories the distance between which is larger than e. If the trajectories are stable according to Lyapounov's criteria, i.e. the close initial representing points do not diverge considerably, then K = 0. 5. Another widely applied criterion for the existence of a state of chaos is Lyapounov's characteristic index A, which also characterizes the average speed of exponential divergence of close trajectories: A = lim

-\n\x(t)\.

t—>oo t

The fact that the parameter A is positive implies that the phase trajectories are diverging, while the value A characterizes the chaos depth.

Chaotic oscillations in radiophysical systems

219

6. Determination of the dispersion D2 = (x — x)2 and plotting of histograms on the basis of process realization. As the strange attractor cuts across a plane known as Poincare plane, the dimension of the phase space can be reduced by unity and a picture of the evolution of the oscillations from periodic to chaotic when changing the controlling parameter (two-dimensional map of the shift) can be obtained on the plane. As an example, we shall present the evolution of the map of the Van-der-Paul equation with nonlinear reactance (also known as Van-der-Paul - Duffing equation): x - e(l - x2)x + x + -yx3 = /9cos(w1<).

Fig. 4.1. Two-dimensional Poincare map given an amplitude increase of an external harmonic signal acting upon a Van-der-Paul generator with nonlinear reactance (Van-der-Paul - Duffing equation)

(4.3)

220

Nonlinear and parametric phenomena: theory and applications

In the case of small values of p (0 < p < 1, 7 at {e,7,u>} = {0.33; 0.7; 2.46}), equation (4.3) may have a periodic or quasi periodic solution. The periodic solution corresponds to a closed limit cycle at the phase plane (x, x). The Poincare section, represented by a stroboscopic sample x(tn) with an interval tn =

is represented

by a single point on the same plane. When there are substantial distinctions between the frequency of the external acting signal ui\ and the frequency of the natural oscillations in the system, wo; almost periodic oscillations are realized x =b\ cost^ii + 62 coso^t , where the frequency u>2 may differ from UJQ as a consequence of the non-isochronous nature of the oscillations. Respectively, the following can be written out for the stroboscopic sample xn = "i + o2 cos

2imu>2

,

. 2TTO

xn = — LO2O2 sin

.

Figure 4.1 shows the Poincare section (two-dimensional map) for the case of almost periodic oscillations and finite n. When n —* 00, the set of points {xn,xn} fill in the closed curve completely. As the amplitude p increases, there is a transition to a seven-stroke cycle (Fig.4.1 b, c), which is stable at 1,9 < p < 2,65. Given a further increase in p, the seven-stroke cycle is disrupted (Fig.4.1 d, e), and in the case of p > 2,75, chaotic oscillations occur. When p = 5, developed chaos can be observed in the system (Fig.4.1 f) 4-1.4- Experimental and numerical methods for investigating chaotic oscillations The experimental investigation of determined and stochastic oscillations in radiophysical systems can be conducted by using standard measuring equipment for analyzing time-dependent, spectral and statistical characteristics of signals [17]. The time-dependent characteristics include the very realization of the oscillation process, the phase portraits of the system obtained by feeding in signals from various points of the system under investigation - Xi, as well as their derivatives, at the inputs of an oscillograph. By choosing different variables {xi,xt} or {xi,ij} different projections of the phase portrait of the system can be obtained. Different orientation of the phase portraits is achieved by turning the respective axes, while adding phase shifts along the coordinates. Data storing oscillographs are particularly useful, when studying complex signals, including chaotic ones. The time-dependent realizations and particularly the phase portraits of the system recorded by means of them not only make it possible to establish the

Chaotic oscillations in radiophysical systems

221

existence of a non-periodic oscillation process, but also reflect the general structure of the attractors emerging in the phase space of the system (e.g. Fig.4.1). For the purpose of obtaining statistical information concerning the motion of a certain dynamic system, such as the distribution function, the correlation function, the power spectrum, etc. standard measures of correlation characteristics, spectrum analyzers, including panoramic spectrum analyzers and the like, can be used. One of the fundamental methods of studying chaotic oscillations in a system with a small number of degrees of freedom is to obtain the Poincare map. Experimentally this is achieved by raising the level of brightness and presenting, on the screen of the data storing oscillograph, only certain points of the phase trajectories {xi,Xj} at preset points of time (Fig.4.2). The brightness of the oscillograph beam is changed by communicating the respective strobing impulses to input z.

Fig. 4.2. Block diagram of set-ups for visualization of Poincare map: a. Map of a radio engineering system with iV-dimensional phase space; b. Map of a non-autonomous radio engineering system In the general case, when investigating a radiophysical system with phase space dimension N, a two-dimensional projection of the Poincare map is obtained as a two-dimensional section of the phase portrait with a hyperplane with dimension

222

Nonlinear and parametric phenomena: theory and applications

N — 1, presented through a linear equation N 1=1

For the purpose of registering the Poincare map, the signals xi, with weight coefficients a; are conveyed to an adder (Fig.4.2a). The moment of the crosssection with the hyperplane is determined by the condition J2aixi = P- At that point of time, a certain voltage comes in from the generator of strobing impulses, which increases the brightness of the respective point of the phase trajectory. A secant hyperplane is selected by choosing the weight coefficients a{ and the level /? of operation of the discriminator. This methodology makes it possible to plot projections of maps corresponding to the maximum values of an arbitrary coordinate - x;. In this case, the discriminator determines the maximum value of the coordinate under the condition i ; = 0, while two other arbitrary coordinates of the process {xj,Xk} are brought up on the oscillograph screen. When studying non-autonomous radiophysical systems, it is convenient for the Poincare map to be presented by those points of the phase trajectory which are separated in time by the period T of the external action (Fig.4.2b). As in the previous case, the brightness of the oscillograph beam is modulated by the generator of a series of strobing impulses with period T. One of the major issues arising in the course of investigating the oscillation systems is connected with the nature of the evolution of the dynamic regimes, when a set (controlling) parameter is changed. Visualization of single-parametric bifurcational diagrams on an oscillograph screen can be achieved, for example, in the experimental set presented in Fig.4.3. When the evolution of the dynamic regime is examined in its dependence on the amplitude of the external harmonic signal, the following steps should be undertaken (Fig.4.3a). The signal X{ is fed at input OY of the oscillograph and saw-tooth voltage is supplied at input OX, which also helps modulate the amplitude of the external harmonic signal. The synchronization of the map on the screen is carried out with the assistance of the generator of strobing impulses in a way analogous to that in Fig.4.2b. Yet another type of bifurcation diagrams comes about, when investigating the frequency properties of the oscillating systems. When visualizing such a type of diagrams, one can use a standard device for analyzing amplitude-frequency characteristics (Fig.4.3b). The bifurcational diagram obtained in this way reflects the response of the system to an external action with a constant amplitude, depending on its frequency. The change in the amplitude of the output signal makes it possible to trace the dynamism in the development of the chaotic regimes. When analyzing the dynamics of nonlinear radiophysical systems, it is hard to give preference to either solely experimental or exclusively numerical methods of investigation. In the course of an experiment, it is simpler and faster to conduct

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223

Fig. 4.3. Block diagram of set-ups for visualization of the evolution of dynamic systems (development of bifurcational diagrams): a. depending on the amplitude of the external harmonic signal, b. depending on the frequency of the external harmonic signal the analysis in the event of variance of the parameters of the system; there is a higher degree of visualization involved in experimental studies, and frequently they are less costly. Yet, when altering the parameters of the system, there is a risk of disrupting the adequacy between the natural and the mathematical model, hence, in all cases it is necessary to control the results numerically, at least with respect to certain typical modes. The latter is also linked with the important circumstance of the dependence, as a matter of principle, of the realizations on the initial conditions. During the experiment, it is by far harder to control the initial conditions of the process under investigation, when a realization may lose stability and be replaced by another one. The experimenter may fail to notice all this and it may lead to wrong conclusions. The numerical investigations of chaos require special algorithms and software, but as a rule they build upon well-known methods of numeric integration of systems of ordinary differential equations. The important aspect with the numeric investigations is the possibility to analyze regular unstable solutions that are not realized in the natural experiment, but whose evolution under changing parameters may result in the occurrence of structurally stable modes, which can be observed experimentally.

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Research practice has shown that the investigation of the dynamics of the radiophysical systems under laboratory conditions is normally conducted by applying the experimental and numeric approaches in parallel. Research software packages have been developed, mainly orientated towards personal computers, which make it possible to analyze the basic characteristics of the radiophysical oscillating systems in both stochastic and determined modes. There are algorithms for calculating the oscillation process, the phase portraits and their projections, the spectral power of the process, the spectrum of the Lyapounov characteristic exponents, the Lyapounov dimension of the attractors, the Poincare map, the singleparameter bifurcational diagrams. The realization of the oscillation process is the simplest, but sufficiently informative characteristic. The Rhunge-Kutta methods of numerical integration of the systems of ordinary differential equations are the most frequently used methods. The phase portraits and their projections provide a visual idea of the stream of trajectories of the system in the phase space and make it possible to identify such important phenomena as the emergence of the limit cycle, the doubling of the period of the limit cycle, the appearance and character of two-dimensional tori, etc. The spectrum of the process power is one of the basic and universal characteristics of the multi-frequency and stochastic oscillations in radiophysical systems. It determines the distribution of the dispersion (the value of the average variance) of the time-dependent realization by frequencies. A quick Fourier transformation followed by an averaging of the set of spectrograms is most frequently used. The spectrum of the Lyapounov characteristic indices and the Lyapounov dimension are among the most important characteristics of the attractor. The Lyapounov indices determine the average stability of the orbit of the attractor in quantitative terms. The concept of Lyapounov dimension is introduced in the following way. Let us arrange the Lyapounov indices as follows: Ai > A2 > ... > Xp. A typical feature of strange attractors is the divergence of the trajectories in accordance with an exponential law. Hence, at least one of the Lyapounov indices is positive. This introduces quantitative understanding in the concept sensitivity to the initial conditions, which exists in the different attempts to define the strange attractor. Currently a Lyapounov dimension is taken to mean a value defined as:

DL=j+\£r\tXi' where j is the maximum integer, for which A + ... + Ay > 0. The correlational dimension along with the Lyapounov dimension is an accessible characteristic from the viewpoint of the volume of computations required to estimate the fractal dimension of the attractor. The correlational dimension

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is estimated on the basis of the correlations between the random points of the attractor. Consider the set of attractor points {x{}, i = 1 , . . . , n, obtained from the time series, i.e. Xi — x(t + ir), r is a constant time interval between the measurements. Due to the exponential divergence of the trajectories, most of the point couples (xi,Xj) at i ^= j will be dynamic uncorrelated couples of essentially random points. These points, however, lie on the attractor. Hence, they will be spatially correlated. This spatial correlation is measured with the help of the integral of the correlation C(l), defined as [17] „,.. _ .. 1 ( number of point couples {i,j} the distance between! n->oo n 2 \ which (\\xi — XJ\\) is less than / J' The link between the correlational integral C(l) and the correlational dimension Dv is subject to the relation C(l) ~ eP", which makes it possible to define Du on the basis of C(l). The Poincare map is a universal informative characteristics of the oscillations in the different dynamic systems, which is convenient thanks to its visual nature and the possibility to identify a broad range of dynamic processes. In order to obtain it, however, one needs a relatively large volume of computer time. The one-parameter bifurcational diagram yields the dependence of the extreme points in the oscillating process for one of the components in the case of an adiabatically slow change in a parameter. These dependencies create possibilities to identify oscillating modes with multiple periods, as well as quasi-periodic, synchronized and chaotic oscillations. A major advantage of this method of identifying oscillating modes is its efficiency from the viewpoint of the necessary computer resources. 4.2. Chaotic oscillations in non-autonomous radiophysical systems with nonlinear reactance and parametric systems A classical paradigm of the Theory of oscillations is the oscillating circuit (linear or nonlinear). The theory of nonlinear resonance (including that exposed in Chapter 3) is usually built on the assumption that a periodic action evokes a periodic response. The new theory of chaotic oscillations precludes this postulate. One of the principles of chaotic oscillations is connected with the appearance of a broad continuous frequency spectrum, when a harmonic signal is used to act on a system. An important class of systems, playing an essential role in radiophysics, is that of the systems of second order, and in particular the non-autonomous oscillating systems with nonlinear reactance. These are mostly amplifiers, converter and generator resonance systems, Josephson superconductive SHF systems and quantum interferometers, semiconductor and ferroelectric resonator systems, parametric resonance filters and the like.

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A major problem when examining the stability of radiophysical systems with respect to the development of chaotic oscillations is the identification of the areas of their emergence in the parameter space of the system. Chaos occurs in nonautonomous radiophysical systems with nonlinear reactance given certain parameter combinations determining the type of the reactive nonlinearity and the nature of the external periodic signal. At this point, we shall focus on the non-autonomous systems most widely used in radiophysics and described by nonlinear differential equations of the second order. When investigating the nonlinear oscillating and generator systems in the previous chapters, we assumed that only strictly periodical oscillations existed in them. It stands to reason that since these systems are described by three independent variables (the third coordinate is time) and since as a rule these systems are nonlinear, chaotic oscillations can be expected to occur in them given certain correlations of the amplitude and frequency of the external force, the damping decrement (growth increment), the coefficient of nonlinearity, etc. Let us consider nonlinear radiophysical oscillating systems influenced by an external harmonic signal, whose motion is described by the following equations [51, 242]: x + 28x + x + x3 = p cos wt, (4.4) x + 26x + x - x3 + jx5 = p cos ut,

(4.5)

x + 26x - x + x3 = p cos ut.

(4.6)

Equations (4.4) - (4.6) represent a very broad class of radiophysical systems, as well as optical and mechanical systems, etc. Equations (4.4) - (4.5) are known in the literature as Duffing equations [51]. Equation (4.6) describes systems with ,,a deep potential well" that have a broad declining sector in the characteristic of the nonlinear reactive element corresponding, for example, to a system with a negative differential capacitance. Fig.4.4 shows characteristics of the nonlinear reactances in equations (4.4 - 4.6). A general picture of the areas of chaotic and periodic motions obtained through a numeric analysis of Eqs.(4.4) - (4.6) is shown in Fig.4.5, Fig.4.6 and Fig.4.7, at 26 = 0,1 and 7 = 0,1. The same figures also show the bifurcational areas characterized by multiplication of the period of oscillations. A complex picture of the location of the different areas can be seen. A more powerful magnification reveals the fine structure of the various areas presented in these figures. Both the chaotic and the periodic areas are sloping to the right, i.e. in the direction of the skeleton curve of the radiophysical system (See sections 3.1 and 3.2). The stationary mode of both periodic and chaotic oscillations is established within an interval in the order of 20 periods of the acting external signal (for Eq.(4.4)), and within approximately 400-600 periods for equations (4.5) and (4.6). The relative share of the plane of parameters {p,a>}, which corresponds to chaotic oscillations, is the smallest in the case of Eq.(4.4): 13%, and the largest in the situation described by Eq.(4.5): 49%. According to this indicator, Eq.(4.6) occupies an intermediate position: 28%.

Chaotic oscillations in radiophysical systems

Fig. 4.4. Plots of the nonlinear reactances in equations (4.4), (4.5), (4.6): 1- y = x + x 3 , 2 - y = x - x3 + 0, lx5, 3 - y = - x + x3

Fig. 4.5. A picture of the position of the areas of chaotic oscillations (double hatching), of periodic oscillations (blank areas), and of periodic oscillations with the respective bifurcation of period multiplication (2, 3, 4, 5 ....) in the plane of parameters {p,u>}. Equation (4.4).

227

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Fig. 4.6. A picture of the position of the areas of chaotic oscillations (double hatching), of periodic oscillations (blank areas), and of periodic oscillations with the respective bifurcation of period multiplication (2, 3,....) in the plane of parameters {P,LO}. Equation (4.5). By way of an example of a non-autonomous resonance system, we shall quote the data of an experimental study of a radiophysical oscillating circuit containing a varactor diode: Fig.4.8a. In line with the presentation in Chapter 3 the processes in the circuit are described by the following system of equations: L— = -Ri-U at

+ Uasinujt

±[C(U)U} = i C(U) = C0(l-aU)-1. After an experimental study of the behaviour of the system, the areas of subharmonic and chaotic response are identified on the plane of parameters { Uo, u>}: Fig.4.8b. The magnified section of the diagram discloses that islands of chaotic behaviour can exist in the plane of parameters {Uo, w}. We shall consider yet another example - a parametric amplifier with a Josephson junction described by the following equation:
(4.7)

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Fig. 4.7. A picture of the position of the areas of chaotic oscillations (double hatching), of periodic oscillations (blank areas), and of periodic oscillations with the respective bifurcation of period multiplication (2, 3,....) in the plane of parameters {p,u>}. Equation (4.6). where tp is the Josephson phase difference, p and u> are the amplitude and frequency of the acting SHF signal, f3c is a parameter equal to the external frequency normalized with respect to the plasma frequency. The numeric analysis of equation (4.7) shows that chaos occurs and the amplification in the parametric amplifier with a Josephson contact is enhanced at an approximately equal correlation of the parameters. This prompts the conclusion that the growing noise observed in almost all experiments with Josephson parametric amplifiers with external pumping is due to the chaotic oscillations occurring in the system. Fig.4.9 reveals the spectrum evolution in the system at /3C = 25. It can be seen (curve 1) that in a mode of periodic oscillations the noise background is 15 orders lower than the desired signal. As the amplitude of external action increases (curve 2) the background noise grows inconsiderably. As chaotic oscillations occur, the background noise soars by some 14 orders (curve 3), so the desired signal exceeds this noise by only 1-2 orders. The areas of periodic and chaotic oscillations in the system depending on the parameters p and w are shown in Fig.4.10.

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Fig. 4.8. Oscillating circuit with a varactor diode (a) and a diagram of the subharmonic and chaotic oscillations in the plane of the parameters {f7o,^o} (b) It is noteworthy that the existence of reactive nonlinearity is a necessary condition for the development of chaotic oscillations in the parametrically excited oscillating systems as well. For example, developed chaos can be observed in the ,,ordinary" nonlinear parametric oscillating circuit described by the equation x + 2Sx + (1 + bcosilt)x + x3 = 0. It is amazing that this relatively simple radiophysical system displays regularities of transition to chaos that are absolutely identical to those in the

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231

Fig. 4.9. Evolution of the spectrum in a parametric amplifier with a Josephson junction; 1,2- periodic oscillation mode, 3 - chaotic oscillation mode

Fig. 4.10. A picture of the position of the areas of chaotic oscillations (double hatching), of periodic oscillations (1), and of periodic oscillations with the respective bifurcation of period multiplication (2, 3,....) in the plane of parameters {p,w}- Equation (4.7). classical Lorentz system describing the processes occurring in a heated volume of liquid by means of equations in partial derivatives. This phenomenon is conditioned

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by the fact that as the bifurcational point of transition to chaos is approached, the multidimensional (and even infinitely dimensional) dissipative system behaves as a small-sized one. This mechanism is another illustration of the unity of the dynamic laws in nature. 4.3. Chaotic oscillations in generator radiophysical systems and SHF short-range self-detecting Doppler radars (autodyne systems) for close radiolocation 4-3.1. Conditions for chaotization of the oscillations in generator systems From the perspective of the new ideas of chaos and stochastization of the oscillations, the following classification of the generator systems widely used in Radiophysics emerges: 1. Radiophysical systems, where oscillations are excited by compensating own losses with negative resistance, or through positive feedback. 2. Resonance radiophysical systems with forced excitation achieved through the action of an amplified signal that has passed through an inertia unit. 3. Resonance radiophysical systems where parametric excitation takes place at the expense of an inertia action of a signal from the resonance system upon elements of this resonance system. There are models of stochastic dynamics in each of the three subclasses. For example, the first subclass can be represented by a modified LC-generator with inertia type of nonlinearity described by the following equations: di L— + Ri + U - Ecosut, at l = z1+l2, ii = C ^ , ii = f(U), at where i, i\, 12 are currents, U is voltage, f(U) is a nonlinear function. The second subclass is represented by a model described by the equations x + 2/3x + x = rj, r) =

fll{x)

and the third one by a mathematical model in the following form: y + 2ey + y3 + zy = 0, z = -eaz + e/3y. The first dynamic system, where chaotic oscillations have been numerically established and investigated, is the following system of Lorentz equations: x = -a(x-y),

y = rx -y - xz, z = -bz + xy.

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233

The Lorentz equations describe a system of the second subclass. The system of Lorentz equations can be presented in a form more relevant to radiophysics as follows: x + 26x + u\x = -ky + f(x, x, y), y + jy - ax + tp(x,x,y), where 6 is the friction coefficient, 7 is an inertia parameter, k and a are coefficients of the linear relation between the variables, / and tp are nonlinear functions containing no linear terms. In this case the variable y has been excluded from the system of Lorentz equations and the following substitutions have been made:

x = x- Vb(^T),

y = °\z-

~X2 - (r -1) (1 - ^ j ,

28 = a + 1, LO2 = b{r - 1), k = ^Jb{r - 1), 7 = 6, a = (2a-b)uj0,

f(x,y)

/3 = ~[-uox

\Z

x2 + — +

l

\ y\x,

J


e > 0,

(this equation was first used to describe the classical valve generator) owing to the lack of reactive nonlinearity. A series of studies demonstrates that the emergence of developed chaos is possible in a system with weak nonlinearity [243]. An interesting point that deserves to be mentioned is the fact that it is possible to suppress the chaotic behaviour of a radiophysical generator by means of external synchronizing action. In this case the strange attractor that describes it is converted to a limit cycle. Furthermore, as different from the phenomenon synchronization of determined generators, when there is no threshold value for its occurrence as regards the intensity of the synchronizing action, in the case of synchronization of chaotic oscillations, the amplitude of the external action should exceed a certain threshold value. In stochastic generator systems, the amplitude threshold of synchronization is directly related to the average positive Lyapounov index characterizing the speed of divergence of closely lying trajectories in the phase space of the system. By

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virtue of this fact, the amplitude threshold of synchronization can serve as a quantitative measure of chaos in the stochastic generator system, which is of particular importance in the experimental investigation of such radiophysical systems. As two generators, respectively of periodic and chaotic oscillations, interact, the possible outcome is either periodization or chaotization of the oscillations. The different modes, all other conditions being equal, depend on the coefficient of the relation between the generators. An interesting point worth mentioning is that the oscillations in a Van-der-Paul generator, which are generally determined in an autonomous mode, can be rendered chaotic given a definite coefficient of a relation with a generator of chaotic oscillations. Interesting phenomena are also observed when two generators of chaotic oscillations interact. This is first and foremost the effect of stochastic synchronization. The latter consists in the following: when a dissipative coupling is established between stochastic generators demonstrating qualitatively different stochastic behaviour, the generators can be synchronized. Given a sufficiently powerful coupling, they start generating chaotic realizations that are identical in their average characteristics (entropy, dimension, spectrum). Moreover, it turns out that even the topology of the strange attractors in the respective partial phase spaces is the same. 4-3.2. Bifurcations and chaos in autonomous and non-autonomous generator systems So far bifurcational phenomena and the conditions for rendering oscillations chaotic have been studied in a large number of diverse radiophysical generator systems operating practically in all the frequency bands available. The main R&D thrust is made into generator systems with a feedback lag, models of a stochastic generator with a tunnel diode, an avelanch-drift diode, bipolar transistors etc., models of a generator with an inertia type of nonlinearity and circular structure, models of a generator with nonlinear resistance [17, 19, 25]. Some of the major methods of analysis were described in the foregoing sections. At this point, we shall use some more typical examples to demonstrate various research approaches. Let us consider the generator of chaotic oscillations shown in Fig.4.11 [20, 25, 67]. The generator is an LC oscillating circuit with a tunnel diode (TD) in a series connection and a negative conductance in a parallel connection, constituting the active element (for example an amplifying triode in the respective regime). The capacitance Ci, in a parallel connection with the TD, upgrades the freedom degree of the system up to 1.5. The oscillating process in the system is described by the following system of equations: x = y-Sz,

y = -x + 2^y + az + /?, pi - x - f(z),

(4.8)

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235

Fig. 4.11. Basic diagram of a generator of chaotic oscillations

f{z) reflects the iV-shaped volt-ampere characteristic (VAC) of the TD in nondimensional parameters, Ug and IQ determine the coordinates of the central point in the falling section of VAC. At fi —1), and quick (along the straight lines x = const and y = const). Given fj. —» 0, the phase space of the system (4.8) degenerates into two semi-planes overlapping in the zone — 1 < x < 1 (Fig.4.12). The transition of the representing point from one of them to the other takes place along the straight lines x = 1 and x = — 1. In line with the strange attractor condition, there are three equilibrium states in the system (one at the origin of the coordinates and two symmetrical to the slow motions plane); all states are unstable, moreover.

Fig. 4.12. Phase space of a generator of chaotic oscillations described by equation (4.8)

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Nonlinear and parametric phenomena: theory and applications

Due to the exponential instability of the system, the deviation of the initial conditions entails both a shift of the unstable foci and an indefinite phase of the transition from one plane to the other, hence, an indefinite number of the oscillations in the voltage trains generated by such a system. Indeed, two arbitrary close phase trajectories around the leap boundary may behave quite differently. The trajectories that remain within the area limited by the tangent to the straight line \x\ = ± 1 , remain in the slow motions plane to perform almost another entire loop unlike those that happen to be beyond that area and, therefore, move to the symmetrical unstable equilibrium of the other plane. It can be shown that the voltage values generated by the system at discrete points of time represent a series of independent random quantities. Since these oscillations occur under the action of non-random energy sources and, within certain finite ranges, are independent of the initial conditions, it is natural that they should be termed stochastic selfoscillations (on the analogy of the regular self-oscillations, whose map is the limit cycle). One of the typical configurations of stochastic generators with a delayed feedback can be presented as a closed circle consisting of a nonlinear element, a linear amplifier, a low-frequency iJC-filter, an ALC-filter. Various nonlinear elements with a falling VAC section are used in the different frequency ranges - SHF devices of the O-type, dinistors, thyristors, unijunction avalanche-and-injection transistors, tunnel diodes, avalanche thermo- and photo-transistors, etc. P—n—p—n structures are particularly promising with respect to the creation of circular generator systems with stochastic behaviour. Fig.4.13a shows the structure of two coplanar field-effect transistors, which, due to its VAC (Fig.4.13b), has been called a A-diode. A p - n - p - n structure (Fig.4.14), where one of the transistors is an MOS-transistor with an induced channel, is used to create a possibility for electronic control of the VAC parameters and for extending the range of the dynamic negative resistance. The structure is used in frequency ranges, where it is an inertiafree element. As the structure is formed in a common semiconductor crystal, it can be applied not only in the low-frequency and radio range, but in the SHF range as well.

Fig. 4.13. Basic diagram of a A-diode (a) and its typical Volt-Ampere characteristic (b) Given two varying parameters in the stochastic generator system with a delayed feedback (e.g. the time constant TRC of the low-frequency i?C-filter and

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Fig. 4.14. Diagram of a nonlinear element (a) and its Volt-Ampere characteristic for different values of the gate voltage Ua: -15 V (1), -12 V (2) and -5 V (3) the constant voltage Ur, applied at the input of the nonlinear element - Fig.4.14a and determining the position of the operating point of the falling VAC section - Fig.4.14b) the result is a two-parameter diagram of the oscillating modes in the generator system - Fig.4.15. The diagram has been obtained by a top-down alteration of the parameter Ur for each fixed value of TRC- There are one-revolution limit cycles in area 1 of the diagram. As the boundary between areas 1 and 3 is transcended, the oscillations lose stability and a two-revolution limit cycle occurs. A transition to this oscillating mode can also be made via area 2, characterized by a definite blurring of the phase portrait of the output one-revolution limit cycle. This blurring is conditioned by the appearance of a very slow modulation of the limit cycle. A further development of the oscillation process in the system, provided that Ur goes down, is related to the loss of stability of the two-revolution limit cycle (passage across the boundary between areas 3 and 4 in the diagram). Area 5 corresponds to four-stroke cycles based on a two-revolution cycle. And finally, an oscillating mode with a strange attractor is realized in the area CA\. A transition from stochastic motion to a three-revolution stable limit cycle takes place at the boundary between areas CA\ and 6. The transition from the three-revolution cycle to chaos CA2, given a further decrease in the parameter Ur, also begins with bifurcation of doubling of the period (area 7). Area p is characterized by stable equilibrium positions. A number of radiophysical and electronic tasks entail the establishment of generator systems with an external harmonic signal acting on them. The nonautonomous generator system also serves as a simple model of generator interaction. This model is used as a basis to study the response, to an input signal, of amplifiers prone to self-oscillation, as well as the processes in superregenerative receiver systems. We have already discussed the classical model of a generator with external harmonic action (4.3) described by a Van-der-Paul's equation with nonlinear reactance. The interest in Van-der-Paul's non-autonomous equation as in a system possible with stochastic behaviour is due to several reasons. Firstly, this equation serves as a model for a large number of phenomena and processes occurring not only in radiophysical systems but in other areas of science and

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Fig. 4.15. Two-parameter diagram of the oscillating modes in a stochastic circular generator system technology as well. Secondly, this very equation has provided the foundation for the development and improvement of the major analytical methods of the Theory of nonlinear oscillations: the asymptotic methods, the small parameter methods, the perturbation methods, etc. However, chaotic oscillations require nonlinear reactance. Hence, the reactive term of the equation is modified to yield an equation of Van-der-Paul - Duffing (4.3). 4-3.3. Creating chaotic oscillations in generator systems with a delayed feedback In this section we shall discuss, from a phenomenological perspective, the creation of stochastic oscillations in a generator with an additional delayed feedback [244]. We shall analyze the mechanisms of chaos appearance both in the case of a constant delay and, in the typical autodyne mode of slow change, when a delay occurs in the additional feedback. The analysis will be conducted on the basis of the following differential equation describing the generator with a delayed feedback from a phenomenological perspective: x-a(lx2)i + Lo2Qx + f(x) = kx[t - r{t)}, (4.9) where f(x) is a nonlinear function, a > 0, k is a coefficient of the additional delayed feedback, r(t) is the delay in the feedback, which can also be time dependent, UJ0 is the ,,linear" part of the own frequency of the generator. Equation (4.9) may also describe a Doppler autodyne system, and in this case 97

2V

CL

CI

r = T0 + at, where r 0 = — , a = —-. Moreover, Eq.(4.9) may characterize one-

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239

mode generation in a gas laser with an additional resonator, one of whose mirrors moves at a definite speed. Equation (4.9) is studied by means of numerical methods at f(x) = Gx3, UJO — 1 and a = 0,1 in two cases: r = To = const and r — To + at. The numerical experiment shows that Eq.(4.9) predetermines stochastic modes in a broad space area of parameters. In this case of stochastic oscillation investigation, the researcher's interest is aroused by the mechanism of chaos generation, given a change in the determining parameters of the system - the feedback coefficient, the nonlinearity coefficient, and the time of the delay. Let us consider the case of an invariable delay (a = 0), which corresponds to a stationary object in the autodyne model (Fig.2.25) and to a stationary mirror in the model of a laser with an additional resonator. We assume G — 0,1 and To = 30. At low feedback factors (0 < k < ko = 0,09) periodic oscillations are excited in the system (4.9). Their frequency / 0 is close to its natural frequency f0 — 0,17 = —. As the feedback factor goes up, the limit cycle of the phase plane, corresponding to the periodic oscillations, loses stability. A two-dimensional torus emerges in the phase space of the system. It is consistent with quasi-periodic oscillations with two basic frequencies: /o and /i = 0,029 ~ — (Fig.4.16a). This torus is stable for values of the feedback factors in the range ko < k < ki, ki = 0,42. As the feedback factor is increased further, two pronounced bifurcations of doubling of the torus quasi-period can be observed (Fig.4.16b): at k = k\ = 0,42 and k = ki = 1,2 respectively. At k = k$ = 1,3 a third, less pronounced bifurcation of doubling of the torus quasi-period can be observed, since the subharmonics are already comparable with the level of the background noise. As the factor k goes further up (k > k± = 1,7), the background noise augments and the individual spectral frequencies (/i, —, —, —,.••) become indiscernible (Fig.4.16c). This 2 4 8 moment can be regarded as the birth moment of chaos. It is worth noting that as the feedback factor k changes, the frequencies /o and /i experience inconsiderable alterations. The dependence of the frequency f\ on the time of the delay To, all other parameters being fixed, is shown in Fig.4.17. The specific values of the feedback factor at which bifurcations of torus birth and doubling of its period occur, depend largely on the value of the constant delay T0 and the nonlinearity G. As the time of the delay increases, the bifurcational values of the feedback delays decrease. Fig.4.18 shows the dependencies of the coefficients fco(l) and ki(2) corresponding to a torus birth and the first bifurcation of doubling on the time delay To (G = 0,1). As the nonlinearity increases, the bifurcational values of the feedback factor also decrease, i.e. chaos occurs at smaller quantities of k. For example, when To = 40, G = 0.1 the following bifurcational values are obtained: ko = 0.03, k\ = 0.3, and

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Nonlinear and parametric phenomena: theory and applications

Fig. 4.16. Spectrum of the process x(t) described by the model equation of a generator with an additional delay feedback (4.9) at k: 0.4 (a), 0.75 (b) and 1.75 (c)

Fig. 4.17. Dependence of the bifurcational frequency on the time of the delay in a generator system with an additional delay feedback ata = 0,G = 0,fc = l when r0 = 40, G — 1, respectively k0 = 0.003, k\ = 0.025 and there is chaos at k > 0.09. The nature of the transition to chaos remains invariable, if the time delay To or the nonlinearity coefficient G is taken as a controlling parameter, rather than k. The difference consists in that fact that as T0 changes, the frequency f\ — — is modified substantially, while a change in G entails mainly an alteration of the frequency f0. For the purpose of establishing the influence of the speed on the reflecting object Eq.(4.9) is examined for a = 0.01. The rest of the parameters are fixed: G = 0.1 and TQ = 20. In this case the transition to chaos is quite different from that

Chaotic oscillations in radiophysical systems

241

Fig. 4.18. Dependence of the bifurcational values of the delay feedback coefficients k0 (1) and ki (2), corresponding to a torus birth and its first bifurcation of doubling, on the time of delay in the a = 0 mode. Even at infinitesimal values of k, quasi-periodic oscillations with two basic frequencies are generated in the system: /o =0.17 and fo = 0.002 ~ af0 (ID - Doppler frequency), i.e. there is a two-dimensional torus in the phase space. As k goes up (k > ki = 0.3), the two-dimensional torus loses stability and a three dimensional torus occurs, i.e. quasi-periodic oscillations with three basic frequencies: /o, fo, and / x = 0.05. The frequency f\ is close to the frequency /i in the case of a = 0. A further increase in the feedback factor boosts the background noise and the result is chaos in the system. The observed transitions to chaos can be characterized by means of the attractor dimension. Here, seeking to assess the dimension of the attractor, we use the correlational index u, which is calculated in the following way. The temporary realization of the process x{t) is used as a basis to establish a sample a;,- = x(t + iT), i = 0 , . . . , N and to build
M(e)

lnC7(£) , vd = rl i m —•

e^°

lne

where C(e) = lim ^ , M(e) is the number of the point pairs (Ci,Cj)> * ¥" j \ the distance between which is less than e. As d increases, initially v
242

Nonlinear and parametric phenomena: theory and applications

Fig. 4.19. Dependence of the dimension of the attractor on the additional feedback coefficient

4.3.4. Chaos in SHF short-range self-detecting Doppler radars Autodyne systems can be defined as a specific class of systems with a delayed self-action. Since, as a rule, these systems operate at short distance, given the different varying conditions in the range, the intensity of the reflected signal can change by several orders. The reflected signals obtained in practice are often very strong, their intensity being comparable to that of the emitted probing SHF electromagnetic oscillations. In this case, typical relaxation instabilities may emerge in the master oscillator of the autodyne system. Under certain conditions they convert it into chaos with a total loss of its informative qualities. At this point we shall investigate, from the perspective of the stochastic conversion of determined oscillations, the possibilities for the occurrence of instabilities of a fundamental nature that can be observed in modern SHF semiconductor autodyne systems with negative resistance [245-248]. The conditions for the emergence of chaos in autodyne systems described by a differential equation of the third order (2.100) are compared with such described by a differential equation of the second order (2.81). For the sake of convenience, we shall hereinafter call them Type I systems and Type II systems. The controlling parameter selected for the investigation is the coefficient of reflection of the signal by a moving object k. The closest reflection of the processes in the real autodyne systems is provided by the differential equations (2.100) and (2.81). To make the numerical analysis more convenient, we present equation (2.100) in the form of the following system of first-order equations:

(4.10)

where x = -^—, y =

'L

, z = ^f, F(i) = /(i)|-G, +G,f(x) + G,f2(x)},

Chaotic oscillations in radiophysical systems

243

xT = x(t - r), zT = z(t - r), f(x) = x(l - dx), (See Figs.2.33 and 2.3). As for the coefficients in system (4.10) and equation (2.81), we take typical values, characterizing respectively the SHF Gunn oscillators and the SHF avalanche-and-drift (IMPATT) oscillators. The time of the delay r is assumed to be r0 = 30. For the purpose of achieving more general analysis, values for k > 1 are also considered. The latter is also of practical significance, bearing in mind the potential of special applications to exert radio counteraction by receiving the probing signal of the autodyne system, amplifying it coherently and emitting it back to it for the purpose of disturbing its operation. The major numeric results are grouped in Table 4.1, which presents the obtained typical bifurcational values of the reflection factor k. Table 4.1 Bifurcational values of the coefficients of reflection from a moving object Koefficients fcj, i = 0,1,2, ...,4

&o

&i

&2

&3

&4

System I

0^07

0^19

0^43

0^47

0^53

System II

0^17

M3

0^97

1/11

M9

/o,/i

/o>/i

^

|- ^

Characteristic spectral

/„

/o,/i

CHAOS

frequencies In this case, the technique of chaos generation is absolutely analogous to that of a generator with an additional delayed feedback, considered above from a phenomenological point of view. Given a relatively small increase in the time of delay to the value To = 40 and all other conditions unchanged, the bifurcational values of the coefficient k change respectively to k£ = 0.015, k[ = 0.11 and k^1 = 0.025, k{T - 0.27. In the case of an increase in the nonlinearity of the function f(x) the bifurcational values of the reflection factor also go down, i.e. chaos occurs at lower values for k. For example, at T0 = 40 and one degree higher values of the coefficients C2 and L2 preceding the cubic term in f(x), the following are obtained: kg = 0.0015, k\ = 0.01 and kl1 = 0.0023, k[* = 0.025. The numeric investigation that has been conducted (Table 4.1) shows that the bifurcational values for k are by far lower for the system (4.10) than for the system (2.81). This is obviously due to the fact that the system (4.10) has an extra half a degree of freedom. From a physical point of view, this phenomenon can be explained by the regenerative action of the direct-current circuit RLC in the system shown in Fig.2.33 which leads to a potential relaxation (bifurcational) instability. For the sake of verifying this conclusion, the system in Fig.2.33 (the system of equations

244

Nonlinear and parametric phenomena: theory and applications

(4.10)) was examined for the same values of the parameters but in the absence of the RiC circuit. In this case the system of equations assumes the form:

!~p-o[i-> + .-!H-& f-a-o/w

(4n)

The results of the numeric experiment indicate that the bifurcational values for k (ki, k2,k$) of the system (4.11) are close to those for a system of type II with no influence of the direct current circuit (k(T^k!,1 jk^1). This confirms the hypothesis of the regenerative influence of the RLC circuit in the system of Fig.2.33 leading to the establishment of more favourable conditions for the occurrence of chaos. The process of chaos occurrence and the gradual disappearance of the Doppler component LOQ from the spectrum of the autodyne system (loss of the informative qualities of the system) was studied through an experiment involving the set-up shown in Fig.4.20, where: AD - an autodyne, PM - a measure of power, RA - a SHF reflective generative amplifier, DL - delay line, DFI - a simulator of a Doppler frequency shift utilizing a mechanically rotating capacitive pin in a waveguide, 5^4 a spectrum analyzer. The time of signal delay is set either by using a high-frequency cable of the respective length or through a couple of receiving and emitting aerials positioned at a certain distance from each other. The damping is offset by the reflective regenerative amplifiers. The experiment with an autodyne system based on a SHF Gunn oscillator and a SHF avalanche-and-drift (IMPATT) oscillator shows that for parameter r = 20e = 10~6 the bifurcational values of the reflection factor at which developed chaos occurs and the Doppler frequency u>r> disappears from the spectrum, are respectively k{ = 0, 35 and k\T = 0,95.

Fig. 4.20. Diagram of an experimental set-up for investigating the chaotic behaviour of a SHF autodyne system The results that have been obtained are of heuristic significance for the more comprehensive characteristics of the behaviour of short-range self-detecting Doppler radars, which constitute a specific class of radio systems of self-action. Taking into account the peculiarities of the operation of these systems at close distances, it

Chaotic oscillations in radiophysical systems

245

follows from the analysis of the numerical experiment that certain conditions (signal delay plus relatively high values of the reflection factor) can be realized, that are sufficient for the occurrence of instabilities as a result of the oscillation chaotization. From the perspective of engineering practice, it should be borne in mind that these instabilities are of fundamental nature and that they are not the outcome of stray feedbacks, defects or other instabilities in the equipment. In certain cases the acquired stochastic behaviour of the oscillations may entail partial or total loss of the informative properties of the radio system.

CHAPTER 5.

ELEMENTS OF RADIOPHYSICAL SYSTEMS

5.1. Generalization of the method of complex amplitudes for linear oscillating systems with periodic parameters and nonlinear systems

A variant of the complex amplitudes method, specially adapted for analyzing linear periodic and nonlinear oscillating systems, is described below (See also 1.2 & 4.3 and [187, 188]). Let us set four different sine functions and construct their transforms in the complex plane through the following correspondence: en = Ai cos(wi* + vi) <- oi = AiejUlt

=

AiejVlejUlt,

a2 = A2 cos(w2t + <^2) ^ a2 = i 2 e ^ 2 ' =

A2ejV2ejU2t,

a3 = A3 cos(oj3t + ip3) <- a3 = A3ejU3t

=

A3e3V3ejoJ3t,

a 4 = Ai cos(w4t + (fii)*-a4=

=

A4e3V<e^*1.

The arrow, correspondence.

A4ejulit

(5.1)

pointing at the original above, stands for the relative

Several mathematical operations typical of the method will be presented. They need no comment: ai

+ 0,2 <— 0,1 + &2i kai <— khi,

dai dai . _ — < — = jwiai,

at

dt

k = const,

f f . hy / aid* *- / aidt = - — .

J

J

(5.2)

J^i

When the classical method of complex amplitudes is used, it is rare to come across products and quotients of harmonic functions, and, in such cases, they are specially regulated (for example, when computing instantaneous powers or modules of the Umov-Pointing vector). Products and quotients of harmonic functions are widely used when analyzing linear periodic and nonlinear systems. That is why their symbols have to be formalized. For this purpose we introduce mathematical operations suitable for such cases: symbolic multiplication "o" and symbolic division " |—", whose meaning 246

Elements of radiophysical systems

247

is revealed by the following examples: a!.a 2 <— ax o a 2 =

a\ + a\ „

a 2 or

hi + a2 „

d .a 2 .a 3 .a 4 <— ax o a 2 o a 3 o a 4 =

ai,

«i + ai a2 + a2 a3 + a3 „

z

z

z

a4

a4 + a4 ai + ai a2 + a2 _ or — ^ — a3) ai

ai

(5.3)

2

— «— [r^ = 7 —ai, a2 02 a 2 + 62

aia 2 a3a4

pi o a2 a 3 o a4

2 2 ai + h\ . a3 + a3 a 4 + a 4 2

Mutually conjugated functions are used here, such as h\ and a^, where aiAjc"'" 1 ' = aie- j ( w i < + ¥ ! l ), Ai = A j e " ^ 1 is a complex conjugate amplitude. It follows from (5.3) that several transforms correspond to the original of the product of harmonic time functions. These transforms differ from each other only with respect to their imaginary parts. It is typical of the symbolic method recounted here that the imaginary functions existing in the intermediate transformations can influence only the imaginary part of the finally achieved transform, which is rejected when the final solution of the problem is obtained. If the original is the sum of harmonic functions, symbolic multiplication and division follow the same rules. For example, if ai 2 = a,\ + a2 and a34 = a3 +
«12 + «i2 . ~ O34,

ai2

,ai2 <— fcr~ = "

2 T^

«12-

2 a34 a34 a34 + a34 The proposed variant of the complex amplitudes method is convenient for studying the processes in linear periodic systems and non-linear systems with polynomial nonlinearity. 5.2. Linear periodic and almost periodic elements of radiophysical systems 5.2.1.Periodic and almost periodic resistance and conductance The objects to be considered are linear temporally periodic and almost periodic resistances and conductance. Let us assume that there is resistance R = Ro[l + mRcos(Slt + tpR)],

(5.4)

248

Nonlinear and parametric phenomena: theory and applications

linked up to an ideal source of harmonic voltage Ug = Ug cos(ujt + ip). When Ohm's law is applied, the following equation is obtained: -Ro[l + mR cos(fi< + tpR)]i = Ug cos(o;t +
(5.5)

In order to make use of the complex amplitudes method, we should set the type of the current as a function of time. It is not difficult to verify that Eq.(5.5) will be correct (with the required correspondence between the frequencies in the left-hand and right-hand parts of the equation) only when the current is set as an infinite series oo

oo

i = Y, !k cos[(w + kfyt + Vk] •- » = Y k~ — oo

h = he'*". (5.6)

ikejiui+kn)t,

k= — oo

By using (5.6) Eq.(5.5) can be ,,moved" in the complex plane following the method of complex amplitudes: oo

Ro^

oo

Y ike*u+lk+1W + Ro Y k= — oo

k= — cx> oo

*

+ i ? 0

he^+W

^

Y

(5.7)

he^+^-^^tl^.

The sign * is used here and further on to denote a complex-conjugate amplitude, for instance m. Any equation of the type of (5.7) is of great importance when analyzing radiophysical systems by applying the complex amplitudes method, so we shall term such equations determining equations. Determining Eq.(5.7) can be presented in the following extended matrix form:

i i 1 r2« [ I

\

I

2

rhR_

2

1

[

^

i i L2 I i i ; ; '

2

I !

'

/ x

; ; _

I I

'

'

;

'

'

(5.8)

!

| : j

™R I

2

\ m n 2

mR I | ^ I j

2 | ;

!

; I I

l

' : . i ; '

'

i

;

l

l

or in brief R I = U.

(5.9)

Elements of radiophysical systems

249

Matrix boxes with zero elements are left empty both here and further on in our presentation. Bold symbols will be used to denote matrixes or vectors, and the dot above will be a reminder that the respective matrix or vector contains complex elements. The matrix in Eq.(5.8) has only three non-zero diagonals, the elements of each of the diagonals are moreover identical. This allows obtaining a result of a unique nature - a precise solution of the infinite system (5.8). Indeed, let us present the system in the following way: 1-2 _ I-i

1

mR

ii -

rhR

h

2

rn-R 1-3

2

1 mR 1-2 '

i

Roio

u"

j_i

(5-10)

~1 + H±*.i=L + ™*k 2

™~R

j0

+

2 j0

ii -

i

^

rhR

1 :

l+!H^h. 2 jfi

mRI3 2 i2

Equations (5.10) yield the following dependence:

rhR L , 2

j

0

=

mR I,

=

mR -4-

2 Jo

(5.11)

m| '

The right-hand side of Eq.(5.11) has turned out to be an infinite continuous fraction with equal elements which is denoted as K. It is known that such a fraction m2 is convergent and has an estimate of |A'| < ——, where mR < 1. Observing that K = - — ~ , we can determine the values of the continuous fraction as: 1+A - 1 ± y/\ - mR '2 = 2 " Some physical considerations obviously require the satisfaction of the condition that given mR = 0, there should be K = 0, which allows of dropping the „-" sign in front of the radical and finally determining: K =

—

—.

250

Nonlinear and parametric phenomena: theory and applications

This result allows obtaining the following expressions as regards the complex amplitudes of all spectral harmonics of the current. f

V*

1

(5.12)

/t=(-i)4-^(

^ . y , .=1,2,...,00.

The vector diagrams of the voltages and currents are plotted in Fig.5.1. The vector diagram in Fig.5.la is trivial for the case when R = const. Fig.5.1b and 5.1c correspond to expressions (5.12). Fig.5.1b shows vectors IQ; I\, I—\\ 1%, 1-2, etc. which turn, in the respective different directions, with frequency Q,. The symmetric character of the diagram in Fig.5.1b allows presenting it in a more compact form - Fig.5.1c - by using the complex function I(t) that changes in time in compliance with a complex periodic law with a period of 2ir/Q,.

Fig. 5.1. A vector diagram of the voltage and currentflowingthrough a resistance connected to a source of voltage: a) a case of constant resistance; b) and c) time-dependent periodic linear resistance with a harmonic variable part Given that the periodic resistance contains two harmonics R(t) = RQ[1 + mi cos(fit + ipi) + m2 cos(2fi< + y>2)],

(5.13)

a matrix equation of the following type can be obtained in an analogical manner

Elements of radiophysical systems

i

; i

m

i

™L

i

1

I ~r \ "hi

^ —

I

mj

!^l

i

m2

~ ™i

i

m

! 1

2 rh2

2 rhx

[ ~2 \

! i /_ ; ! ;

•

-

;

!

i

i f

i

i

i

'

I > -1 • i i % ;. I /0 1=1 ilg I (5-14) mi

—

2

\

[

2

T ~

! ; !

I ^_ 1

2

251

~Y [

i

! ;

'

•'

'

i

i

i

i

,

; \

; \

; • \ •2 i

| \

; \

\ :

ii

:

i

If the periodic resistance is expressed by k harmonics, then the respective matrix R will contain 2k+ 1 non-zero diagonals situated symmetrically with respect to the main diagonal. In the general case of a periodic function R(t), represented by an infinite harmonic series, matrix R will not contain zero elements. The matter is much more complex when the function is almost periodic. In this case instead of (5.14), the resulting systems are multidimensional and infinite, and they are not specifically studied by the Theory of oscillations. The multidimensional systems of algebraic equations below are used for classifying the respective radiophysical systems. Let us assume that resistance changes in time according to the law R(t) = RQ[1 + mi cos(fti< +
(5-15)

where fii and SI2 are incommensurable frequencies, i.e. R(t) is an almost periodic function, whose branching is equal to two (we introduce the concept of ,,branching", which is related to the number of incommensurable frequencies; obviously, a periodic function has a branching equal to one). The equation concerning the current flowing through the resistance, given an ideal source of voltage, can be written down in a form analogical to that of (5.5): Ro[l + mi(fl]t + ifi) +m,2 cos(fM + tP2)]i = Ug cosuit.

(5.16)

The analysis indicates that Eq.(5.16) will be correct only if the current is given in the form of a double sum i=

00

00

^2

^2

Ikicos[(u) + fcfii + l&2)t +
k=—00 / = —00 00

*-"i=

E

k=~OO

00

E

I *' c '' ( ' iH "* ni+ma) ''

I—— OO

ikl =

hie^'.

252

Nonlinear and parametric phenomena: theory and applications In this case the determining equation is obtained in the form oo

*o E

/ .

oo

.

*

*

\

E (^,«-i + ^ - i , . + i , i + ^ , . + ^ , i + i )

k= — oo(= — oo \

xej(w+kn1+in2)t

=

/

(5.17)

jjgeiut.

When the components of equal frequencies are selected, an infinite linear algebraic system of an extraordinary form is obtained. Since we are not in a position to write it out in full, we shall confine ourselves to the following short presentation: R..i = U.

(5.18)

The two horizontal dots in (5.18) denote a peculiar double-index multiplication of matrixes; I and U are infinite square matrixes with respective indices (k, 1) and (i,j) and with the following structures:

__1

1

1^-2,-2^-2,-1 1-2,0 1-2,1 I-2,2\ k

i-f-1,-2-^-1,-1 I—lfi

I = ; /o,-2 ^0,-1

^0,0

-f-1,1 -^-1,2;

j

j

j ,j

I

^0,1 h,2 | , U = |

Ug

| ,

I h,-2 h,-i h,o h,i

h,2 ;

I

I

j ^2,-2 h,-i

h,2 ;

I

;

h,o

h,\

H—{Ritj:k,i} is a four-dimensional infinite matrix, z, j ,fc,/= — o o , . . . , — 1 , 0 , 1 , . . . , oo. In the case under consideration, matrix R. has proved to be quite sparse, since its non-zero elements are only the following ones: Rk,l,k,l = Ro, Rk,i,k-i,i = ~miRo,

Rk,i,k,i-i = ^rh2Ro,

(5.19)

Rk,i,k+i,i = 2 m i-Ro, Rk,i,k,i+i = -m2RoLet us explain the product of matrixes Ft, I, U in (5.18) as compared to the ordinary product of matrixes. Let us assume that we have three ordinary (twodimensional) rectangular matrixes: A={oi,j}, B={6iit},

i = l,2,...,p k = l,2,...,r,

,

j = l,2,...,q, C = {<:,•,*}.

Elements of radiophysical systems

253

If these matrixes are related through the ordinary product A.B = C, this implies that cik = YJ)=I aijbjkThe ordinary product of matrices can be termed single-index product since it leads to a summing by a single index (in this case by j). Let us assume now that the first matrix is four-dimensional, i.e. A = {aStllilik} , e = l , 2 , . . . , s , /* = 1,2,... ,t, B = {&,-,*}, C = {<:«,„}. In this case the relation between the matrices is realized through a peculiar product A • »B = C, where p

i

«=i 7=1

Obviously this product differs from the ordinary one by the existence of a double sum. It is natural to qualify such multiplication as a two-index one. If another harmonic component with incommensurable frequency is added to (5.15), the equation complying to the generalized Ohm's law will acquire the following form R . . . I = U,

(5.20)

where three-index multiplication appears. Moreover, the matrix of the system is six-dimensional, while the other matrixes are three-dimensional. Following this logic, a whole family of multidimensional matrixes can be introduced: two-index, three-index, four-index ones, etc. They are all of practical importance. As a rule, the matrix of the system has an even number of dimensions, while the other matrixes have half the dimensions of the system one. Let us now consider the other typical problem - the variable resistance is attached to an ideal source of harmonic current ig = Ig cos(a;£ +
(5.21)

The same two cases tackled above will be considered here (See (5.13) and (5.15)): 1/ R = R0[l + m-i. cos(flt + ipi) + m2 cos(2fi< +
+ ip{) + m2 cos(9,2t + ^2)]-

In the first case the voltage symbol is obtained in the form of a finite sum U=

E

k=~2

t/*e^+ M )< ,

Ui = \rhxRoig 1

*

,

(5-22)

U2 = lrh2R0ig,

•

U-i = -miRoIg

U0=R0Ig,

•

,

U-2 =

1 *

•

-m2R0Ig.

254

Nonlinear and parametric phenomena: theory and applications

This sum can be regarded as an almost periodic time function with a branching equal to one. In the second case we obtain:

U= E k--\

E Uk,eX"+ka*+m*

J=-l

Ui,o = -miRoig ^-1,0 = -^Rgig

,

, Uoo = R0Ig, Uo,i = -m2R0ig,

,

)7o,_i =

(5-23)

-rn,2R0ig.

The sum written down in (5.21) is an almost periodic function with a branching equal to two. Obviously the considered problem with supply from a source of current is incomparably simpler than that with supply from a source of voltage. It is not difficult to be convinced that the problems related to the investigation of the properties of periodic or almost periodic conductance are analogical to the problems related to periodic and almost periodic resistance considered above. Indeed, if a linear periodic conductance, presented as G = Go[l + mGcos(£lt + ipG)],

(5.24)

is linked up to an ideal voltage source Ug cos(cot + tp), then the determining equation will be i = G 0 [l + mG cos(fii +
(5.26)

Comparing (5.25) and (5.26) with (5.21) and (5.5), we become convinced that these equations are equivalent to an approximation of the denotations. In practice it may be relevant to make use of the choice of a source (current or voltage) and of the value (resistance or conductance) for simplifying the analysis. 5.2.2. Periodic inductance and magnetic susceptibility When a periodic inductance of the type L = Lo[l + mLcos(Q,t +
(5.27)

is attached to an ideal source of harmonic voltage Ug, the following determining equation is obtained oo k= — oo

oo k= — oo

Elements of radiophysical systems *

oo

+ju>L0?± T 1

255

h+^+k^

(5.28)

= Uge^.

k=-oo

This equation is analogous to Eq.(5.7) - the difference is that here multiplier .Ro is replaced with JUILQ. Hence, all conclusions concerning (5.8) will be applicable here as well. In this case, the vector diagram differs from that in Fig.5.1 only in that vectors IQ and Ug are moved apart from each other at an angle of —. When the periodic inductance is connected to an ideal source of harmonic current ig = Ig cos(art +
, Uo = juLoIg, (5.29)

U! = j(u + fl)£o^/9 , ZJ

U-i = j(w -

n)L0^ig. A,

This case is analogous to the one considered above, where a periodic resistance is connected to a source of current, with the important difference that here U-i ^ U\. This difference can be visualized in the vector diagram - Fig.5.2, plotted in correspondence with (5.29). It is obvious that the instantaneous complex voltage amplitude U(t) leads the complex current amplitude with an angle, which is periodically either larger or smaller than 90°, but its average is equal to 90°. If vector Ig is regarded as motionless, then vector U(t) will periodically change its module and direction with angular frequency ft, constantly remaining in a definite sector wide la (See Fig.5.2b). By way of visualization, one can imagine that U{t) is a harmonic oscillation modulated both by amplitude and by phase with a modulation frequency ft. As a rule, the ,,modulation frequency" ft in modulationparametric systems is much higher than the ,,carrier" frequency w, i.e. ft ^> u>.

Fig. 5.2. A vector diagram of the voltage of a linear periodic inductance with a harmonic variable part connected to a source of harmonic current: a) in full form, b) in compact form The considered case of connection of periodic inductance to a source of current is characterized by this specific peculiarity - the appearance of phase modulation. Such phase modulation fails to appear when a periodic inductance is connected to a source of ideally harmonic voltage.

256

Nonlinear and parametric phenomena: theory and applications

If the inductance changes in time in accordance with an almost periodic law, then the modulation that comes about is characterized by an almost periodic law and a multidimensional quality emerges. Then sector 2a is defined in multidimensional space. Consider a periodic magnetic susceptibility (a value opposite to inductance) F — Fo[l + mrcos(nt + ipr)]- In the case of a connection to an ideal source of harmonic voltage Ug, the following complex form of an expression is obtained for the flowing current

i = L^-v* T _ roUg This case difference that When the current ig, the

_T

+ ioe^ + iie^+v\ f _ rhr V0Ut

•

_ mr T0Ug

(5.30)

is close to the one considered above (See (5.29)), with the onlyno phase modulation emerges here. magnetic susceptibility is linked up to an ideal source of harmonic following determining equation is obtained

V" [

Uk

xej(u+kn)t^

I ^r

^k~l

™T ^k+1 1

jgeiut^

( 5 3 1 )

which describes an infinite system of the type

: i I

i iw-n ;;

™L

2

-JTO i ^ i ik 11 '

m r

~2~

i I

I

I

x

'

'

I

I

ii

a,

iil,i i i ; j •' j

i M=nn. i

i

1

L

_

i

i

i

J

L

J

Lo + n\ ; Ui. I i

(5-32)

I

or in brief: f (jw)" 1 !! = I, where u — diag(..., w — kil,... ,w - fi, w, w + Q,..., u + kQ,,...) is an infinite diagonal matrix of frequencies. The emergence of a matrix of frequencies is indicative of the existence of phase modulation. Phase modulation appears when the inductance or magnetic susceptibility are tied to a source of harmonic current and fails to appear when they are connected to a source of harmonic voltage. This is obvious, since in the first case an infinite sum of voltage harmonics appears and each term is divided into its frequency with the respective phase shift. As a result, a matrix frequency multiplier appears which conditions an angular modulation.

Elements of radiophysical systems

257

5.2.3. Periodic capacitance and electric elastance When a variable capacitance C(t) = CQ[1 + me cos(fi£ +
i = i-ns*"-0*

+ ioejult + iie^+V\

Io = JLOCOU,,,

U = j(w - n ) c o ^ u g .

h = j{w + n)co^u9,

(5 ' 33)

The vector diagram for this case is presented in Fig.5.3. This case is dual to the case of inductance with a source of current (Fig.5.2).

Fig. 5.3. A vector diagram of the voltage and currents flowing through periodic capacitance with a harmonic variable part connected to a source of harmonic voltage: a) in full form, b) in compact form When the capacitance is connected to an ideal source of harmonic current, the following determining equation is obtained oo

/ .

;<"Co E

*

y t , + ^ + ylJW

k= — oo \

\

e* +ffi "=V'-

(5.34)

/

which structurally coincides with (5.7), given the substitution jwCo-^>R,'mc-*mRWhen a periodic electric elastance S(t) = SQ[1 -f ms cos(Q,t -f- Vs)] ( a quantity opposite to capacitance) is connected up to an ideal source of harmonic voltage, the result is a determining equation oo

_ •s 3

°u

f

S^ \

f

\oJ + kn+

. ms

j

2 uj + (k-l)Q+

* m s

"|

f +1

2 uj + (k + l)Q\

x c ^ + * " ) * = U-,e^ t , (5.35) which differs from (5.31) only by the denotations. The respective infinite system is of the type

258

Nonlinear and parametric phenomena: theory and applications

r

1

: I j

!Zli _2

I

-js0 i ^ : 2 I i

1

i!

*

i

1

I

I

i

r

' I

: './_!'' : \j • '<

* _ _

I

r

I !

: iw-ft | ; _ _

!^ ! I 2 :': rhs j| ~9~ ii

I

"i

!

i

' ;

!

\ io = u, ;j | | j ' ii• i ! i w + ^ i i -fI i ; i I

L

J

L

^ 5 - 36 )

J

or in brief: S^'w)" 1 ! = U. Given a linkage of the electric elastance to an ideal source of harmonic current, the result is

U = E Uke*»+k0» , Uo = ^ , (5-37) jT

rhs So •

•

_msSOf

The general conclusion is that the capacitance and the electric elastance linked up to an ideal source of harmonic voltage yield a current modulated by amplitude and phase, while their connection to an ideal source of harmonic current results only in amplitude modulated voltage. 5.2-4- Power consumed by periodic elements of the radio circuits An instantaneous power consumed by a periodic resistance connected to an ideal source of harmonic voltage is represented as 7T

p=Ugl^P=El±Ei.l

I 7T

1

=

°°

L £

k= — oo

1

°°

uaJtCi(2uH-*n)* + 1 £

u,ike*a*. (5.38)

k= — oo

In the special case when one can find such k = ka < 0, where the equation 2w + kail = 0 would be fair, the average power is equal to

Pav = Re (^

+^

j = \u9h + \uglka cos(^ +
(5.39)

Fig.5.4 shows a typical vector diagram of the symbol of instantaneous power p plotted on the basis of expression (5.38), as well as the qualitative time dependence of the instantaneous power p(t) consumed by a periodic resistance.

Elements of radiophysical systems

259

Fig. 5.4. Illustration of the instantaneous power consumed by periodic resistance connected to an ideal source of harmonic voltage: a) A typical vector diagram of the symbol of the instantaneous power; b) Quality dependence on the time of the instantaneous power The vector diagram of the symbol of instantaneous power in dynamics (Fig.5.4a) can be explained by using the vector diagram in Fig.5.1c. It follows from (5.38) that the ,,lying" vector

1

*=-oo

is always situated in the direction of the real axis and that it undergoes modular changes with frequency Q, according to a periodic law, and its value cannot be negative. The ,,rotating" vector

Pr = 1 J2

U9he^+k^

Jb=-oo

turns regularly, with an angular speed 2a;, round the end of vector P; undergoing modular changes analogous to vector P/. Fig.5.4b is a consequence of Fig.5.4a. The almost periodic function p(t) is built on the basis of two frequencies: frequency Q,, at which modules Pi and Pr change in synchronism, and frequency u>, when vector Pr turns round the end of vector Pi with angular speed 2a;. Given fixed frequencies u> and il, the form of function P(£) may be changed in a wide range by changing the initial phases